Explanation Procedure used: The standard form of the linear differential equation: d y d x + P ( x ) y = f ( x ) The integrating factor for above equation is e ∫ P ( x ) d x . The product rule of integrals is, ∫ u d v = u v − ∫ v d u Calculation: Consider the given differential equation. y ′ = 2 y + x 2 + 5 Rewrite the above equation in standard form. y ′ − 2 y = x 2 + 5 (1) Compare the above equation with standard equation. P ( x ) = − 2 and f ( x ) = x 2 + 5 Calculate the integrating factor for the equation. e ∫ P ( x ) d x = e ∫ − 2 d x = e − 2 x Multiply equation (1) with integrating factor. e − 2 x y ′ − 2 e − 2 x y = e − 2 x x 2 + 5 e − 2 x d d x [ e − 2 x y ] = e − 2 x x 2 + 5 e − 2 x Integrate on both sides of the equation. ∫ d d x [ e − 2 x y ] d x = ∫ ( e − 2 x x 2 + 5 e − 2 x ) d x e − 2 x y = ∫ e − 2 x x 2 d x − 5 2 e − 2 x + C (2) Solve the integral ∫ e − 2 x x 2 d x by product rule. Consider u = x 2 and d v = e − 2 x So, d u = 2 x d x and v = − 1 2 e − 2 x Substitute above values and solve the integral ∫ e − 2 x x 2 d x by product rule

6th Edition

ISBN: 9781284127003

Author: ZILL

Publisher: JONES+BARTLETT LEARNING,LLC (CC)

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

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Chapter 2.3, Problem 8E

To determine

The general solution of differential equation, largest interval over which solution is defined and transient term in the general solution of differential equation.

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The general solution of differential equation, largest interval over which solution is defined and transient term in the general solution of differential equation.

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The general solution of differential equation, largest interval over which solution is defined and transient term in the general solution of differential equation.

Chapter 2 Solutions