A First Course in Differential Equations with Modeling Applications (11th Edition)

Chapter 7

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; The function is defined as Apply the ... more

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; The function is defined as Apply the ... more

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; The function is defined as Apply the ... more

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; The function is defined as Apply the ... more

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; The function is defined as Apply the ... more

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; The function is defined as Apply the ... more

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  ; From the graph, compute the equation of for ... more

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; From the graph, compute the equation of for ... more

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; From the graph, compute the equation of for ... more

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; From the graph, the equation of for the ... more

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  ;  The Laplace transform for a function for is... more

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; Apply the definition of Laplace transforms to ... more

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; The function is defined as .Apply the ... more

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; Apply the definition of Laplace transform to ... more

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; The function is defined as .Apply the ... more

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; The function is defined as .Apply the ... more

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; The Laplace transform of a function is defined... more

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; Substitute for in and simplify the obtained ... more

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; The Laplace transform of the function is ... more

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; Substitute for in and simplify the obtained ... more

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; Substitute for in and simplify the obtained ... more

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; Substitute for in and simplify the obtained ... more

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; Substitute for in and simplify the obtained ... more

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; Simplify the expression for using the formula... more

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; Simplify the expression for using the formula... more

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; Substitute for in and simplify the obtained ... more

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; Simplify the expression for using the formula... more

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; The function is equivalent to .Substitute for... more

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; Simplify the equation using the formula; .  ; ... more

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; Substitute for into and simplify the ... more

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; Substitute for in and simplify the ... more

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; Substitute for into and simplify the ... more

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; Substitute for into and simplify the ... more

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Substitute for in and integrate the ... more

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The Laplace transform of a function is defined as... more

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; Use the fact that for , and substitute for ... more

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; Use the fact that for , and substitute for ... more

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; Use the fact that for and substitute for ... more

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; The Laplace transform of the functionis ... more

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If ; The Laplace transform of a function, is ... more

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The inequality implies that as the exponential ... more

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Evaluate using the formula, .Write the Laplace ... more

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For to be a linear transform, must be equal to ... more

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The function is not continuous in the last ... more

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Write the Laplace transform of the function ,  as ... more

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The Laplace transform .Evaluate the integral ... more

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Write the left-hand side of where is a constant ... more

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; Use the change of scale theorem, that is, if ... more

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; Use the change of scale theorem, that is, if ... more

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; Use the change of scale theorem, that is, if  ... more

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; Use the change of scale theorem, that is, if ... more

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; Compare the inverse Laplace transform with, to... more

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  ; Compare the inverse Laplace transform with, ... more

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; The inverse laplace transform is defined as .... more

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; Use the formula, for the expansion of the ... more

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; Use formula, for the expansion of the function... more

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; Use the formula, for the expansion of the ... more

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; The inverse laplace function is defined as .Use... more

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; The inverse laplace transform is given as .... more

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; The inverse Laplace transform is given as , ... more

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; The inverse Laplace transform is given as , ... more

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; The inverse laplace transform is given as .... more

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; The inverse laplace transform is given as .... more

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  ; Factor out four from the numerator and ... more

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; Factor out four from the denominator of the ... more

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  ; Rewrite the function, by using the linear ... more

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; Rewrite the function, by writing the ... more

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  ; Rewrite the function, in the form of partial ... more

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; Rewrite the function, in the form of partial ... more

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  ; Factor the denominator of the fraction, using... more

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; Factor the denominator of the fraction, using ... more

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; Rewrite the fraction, in the form of partial ... more

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  ; Use the formula, to de-factor the denominator... more

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; Write the partial fraction decomposition of the... more

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; Write the partial fraction decomposition of the... more

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; Take out common from the denominator of the ... more

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; Decompose the fraction into partial fractions ... more

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; Write the partial fraction decomposition of the... more

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; Factor the denominator of the fraction, then ... more

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; Decompose the fraction into partial fractions ... more

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; Factor the denominator of the fraction, then ... more

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; Find the laplace of the function by using the ... more

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; Substitute for into and simplify the ... more

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; Substitute for into and simplify the ... more

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; Apply the laplace transform on both sides and ... more

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; Apply the laplace transform on both sides and ... more

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; Write as in the expression.Apply the laplace ... more

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; Apply the laplace transform on both sides and ... more

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; Apply the laplace transform on both sides and ... more

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; Apply the laplace transform on both sides and ... more

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; Apply the laplace transform on both sides and ... more

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; Apply the laplace transform on both sides and ... more

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; Apply the laplace transform on both sides and ... more

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; Apply the laplace transform on both sides and ... more

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; Apply the laplace transform on both sides and ... more

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; Apply the laplace transform on both sides and ... more

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; Apply the laplace transform on both sides and ... more

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; Let one of the functions be .The Laplace ... more

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; Let and take the Laplace transform of the ... more

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It is enough to show that to prove that .Apply ... more

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; Use the first translation theorem, that is, if... more

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; Use the first translation theorem, that is, if... more

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; Use the first translation theorem, that is, if... more

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; Use the first translation theorem, that is, if... more

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; Rewrite the Laplace transform, by simplifying... more

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; Rewrite the Laplace transform, by simplifying... more

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; Use the first translation theorem, that is, if... more

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; Use the first translation theorem, that is, if... more

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; Rewrite the Laplace transform, using the ... more

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; Rewrite the Laplace transform, using the ... more

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; Use the fact that is shifted two units to the... more

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; Use the fact that is shifted one unit to the ... more

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; Rewrite the fraction, by completing the square... more

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  ; Rewrite the fraction by completing the square... more

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; Rewrite the fraction by completing the square ... more

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; Rewrite the fraction by completing the square ... more

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; Fix the numerator of the fraction so that this... more

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; Fix the numerator of the fraction so that this... more

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; Rewrite the fraction in the form of partial ... more

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; Rewrite the fraction in the form of partial ... more

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; Apply laplace on both sides of the given ... more

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; Apply laplace on both sides of the given ... more

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; Apply Laplace transform on both sides of the ... more

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; Apply the Laplace transform at both sides of ... more

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; Apply the Laplace transform at both sides and ... more

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; Apply the Laplace transform on both sides and ... more

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; Apply the Laplace transform at both sides and ... more

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; Apply the Laplace transform at both sides and ... more

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; Apply the Laplace transform at both sides and ... more

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; The differential equation is with initial ... more

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; Take the Laplace transforms on both sides of ... more

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; Take the Laplace transforms on both sides of ... more

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; The force equation for the Hooke’s law is , ... more

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; ; The differential equation for the ... more

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Take the Laplace transform of the equation and ... more

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; The differential equation of an series circuit... more

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; On comparing with the left side of the formula... more

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; Compare the function with then and .Determine... more

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; On comparing with the left side of the formula... more

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; Compare with the formula, then .Evaluate the ... more

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; Compare with formula , it can be observed that... more

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; Compare the expression with the formula,, it ... more

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; Compare the expression with , it can be ... more

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; Expand the expression using the formula and ... more

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; Compare the expression with , it can be ... more

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; Compare the expression with , it can be ... more

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; Use the inverse form of second translation ... more

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; Use the inverse form of second translation ... more

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It can be seen from the graph that for , the graph... more

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Unit step function: Laplace transform: ; Reduce ... more

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Unit Step function: Laplace transform: ; Find the ... more

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Unit Step function: Laplace transform function: ; ... more

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Unit Step function: Laplace transform: ; Find the ... more

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Unit Step function: Laplace transform: ; Find the ... more

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Unit Step function: Laplace Transform: ; Find the ... more

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Unit step function: Laplace transform: ; For non ... more

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; Take laplace on both sides of the equation, ... more

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; Take laplace on both sides of the equation, .... more

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; Take laplace on both sides of the equation, .... more

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; Take laplace on both sides of the equation, .... more

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; Take laplace on both sides of the equation, .... more

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; Take laplace on both sides of the equation, .... more

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; Take laplace on both sides of the equation, .... more

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; Take laplace on both sides of the equation, .... more

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Equation of motion of string:Graph of the equation... more

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; In a spring-mass system, mass weighing ... more

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; From the graph, it is observed that .Find the ... more

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; From the graph, it is observed that .Find the ... more

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; The static deflection of a uniform beam of ... more

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; The static deflection of a uniform beam of ... more

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; The static deflection of a uniform beam of ... more

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; The static deflection of a uniform beam of ... more

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; The derivatives of transforms theorem states ... more

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; The derivatives of transforms theorem states ... more

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; The derivatives of transforms theorem states ... more

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; The derivatives of transforms theorem states ... more

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; Apply the derivatives of transforms theorem ... more

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; Apply the derivatives of transforms theorem ... more

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; Apply the derivatives of transforms theorem ... more

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; Apply the derivatives of  transforms theorem ... more

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; Take the Laplace transform of the differential ... more

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; Take the Laplace transform of the differential ... more

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; Take the Laplace transform of the differential ... more

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; Take the Laplace transform of the differential ... more

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; Write the function in terms of the unit step ... more

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; Write the function in terms of the unit step ... more

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Enter the equation in the graphing utility to ... more

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Enter the equation in the graphing utility to ... more

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; Take the Laplace transform of the differential ... more

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; Take the Laplace transform of the differential ... more

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, ; Apply the definition of convolution of two ... more

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, ; Apply the definition of convolution of two ... more

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, ; Apply the definition of convolution of two ... more

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, ; Apply the definition of convolution of two ... more

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; Apply the convolution theorem to according to ... more

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; Apply the convolution theorem to according to ... more

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; Apply the convolution theorem to according to ... more

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; Apply the convolution theorem to according to ... more

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; Compare the integral with the definite ... more

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; Rewrite the integral and compare it with the ... more

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; Compare the integral compare it with the ... more

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; Compare the integral with the definite ... more

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; Compare the integral with the definite ... more

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; Compare the integral with the definite ... more

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; Use the derivative of transform formula to ... more

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; Use the derivative of transform formula to ... more

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  ;  The inverse form of transform of an integral ... more

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;  Use the inverse form of convolution theorem ... more

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; Use the inverse form of convolution theorem ... more

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;  The inverse form of transform of an integral ... more

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; Apply laplace transform on both sides of the ... more

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; Apply laplace transform on both sides of the ... more

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; Apply laplace transform on both sides of the ... more

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; Apply laplace transform on both sides of the ... more

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; Use the laplace transform on both sides of the ... more

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Let the Laplace transform of f(t) be L(f(t))→F(s).... more

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; Use the Laplace transform on both sides of the ... more

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; Determine the equation of graph for by using ... more

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To solve equation (15) subject to i(0) = 0 with ... more

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HERE YOUR ANSWER... more

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; Substitute 1 for , 2 for , 1 for into the ... more

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; Bessel’s equation of order  is .Take laplace ... more

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; The differential equation is given by .Take ... more

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; Rewrite the given function by moving inside ... more

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; Let .Take laplace of the equation, missing ... more

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; The given initial value problem is with ... more

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; The given initial value problem is with ... more

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; The given initial value problem is with initial... more

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; Take Laplace Transform on both sides of the ... more

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; Take Laplace Transform on both sides of the ... more

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; Take Laplace Transform on both sides of the ... more

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; Take Laplace Transform on both sides of the ... more

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; Take the Laplace transform on both sides of the... more

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; Take the Laplace transform on both sides of the... more

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; Take the Laplace transform on both sides of the... more

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; Take the Laplace transform on both sides of the... more

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; Take the Laplace transform on both sides of the... more

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; Take the Laplace transform on both sides of the... more

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; Take the Laplace transform on both sides of the... more

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; Take the Laplace transform on both sides of the... more

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; Take the Laplace transform on both sides of the... more

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for , and for ; Take the Laplace transform on ... more

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; Take the Laplace transform on both sides of the... more

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Yes, the solutions of both the initial value ... more

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; Take the Laplace transform on both sides of the... more

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; Apply the laplace transform to the differential... more

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; Apply the laplace transform to the differential... more

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Step 1: Step 2: Step 3: Step 4: more

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; Apply the laplace transform to the differential... more

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; Apply the laplace transform to the differential... more

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; Apply the laplace transform to the differential... more

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; Apply the laplace transform to the differential... more

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; Add and subtract the equations and to ... more

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; Take laplace transform on both sides of the ... more

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; Take laplace transform on both sides of the ... more

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; Take laplace transform on both sides of the ... more

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; Substitute 3 for ,2 for , 1 for and 1 for in ... more

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; Net force acting on the masses $ and are and... more

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; Substitute 6 for ,5 for , 1 for ,1 for , for ... more

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; Substitute 50 for , for and 0 for and in ... more

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; Substitute 2 for , 60 for , 50 for , for , ... more

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False ; Let the function is which is not ... more

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False ; Rewrite the function in the exponential ... more

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; Let .Find the Laplace transformation of the ... more

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; Rewrite the function .Here, .Find the Laplace ... more

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; Find the inverse Laplace of the function... more

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; Find the inverse Laplace of the function... more

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Approach to solving the question: Detailed ... more

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; Simplify the expression to obtain a sum of ... more

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; Compare with . So, and .To evaluate inverse ... more

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; The expression can be written as,  Compare ... more

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; The expression can be written as  Compare ... more

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-5 ; Compare with . So, .As , so and is of ... more

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; Compare with . So, and .As , so .As , so... more

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; The expression can be written as,As , so .As... more

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and ; The expression can be written as,... more

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; The equation that satisfies the given graph ... more

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; The equation that satisfies the given graph ... more

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; The equation that satisfies the given graph ... more

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; The equation that satisfies the given graph ... more

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; ; ; The equation that satisfies the graph is ... more

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  ; The equation that satisfies the graph is and ... more

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  ; The equation that satisfies the given graph ... more

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; Take the laplace transform on both side of the ... more

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; Take the laplace transform on both side of the ... more

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; Substitute for , for , for , 0 for , and 5 ... more

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; Apply the Laplace on both sides of the equation... more

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; The given differential equation is .Apply ... more

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; For r  can be expressed as .Therefore, a given ... more

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; For a function, This can be expressed as .The ... more

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; The given differential equation can be written ... more

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; The given differential equation is .Apply the ... more

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; The solution of an integro-differential ... more

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; The given differential equations are and . ... more

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; The given differential equations are and . ... more

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; It is given that, , , and . The integral ... more

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; The EMF equation can be expressed as .Apply ... more

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; The equation for the load per unit length is ... more

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