Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 10 - Introduction to Differential Equations - 10.4 First-Order Linear Equations - Exercises - Page 524: 8



Work Step by Step

This is a linear equation and has the integrating factor as follows $$\alpha(x)= e^{\int P(x)dx}=e^{\int x dx}=e^{\frac{x^2}{2}} $$ Now the general solution is \begin{align} y&=\alpha^{-1}(x)\left( \int \alpha(x)Q(x)dx +C\right)\\ &=e^{-\frac{x^2}{2}}\left( \int xe^{\frac{x^2}{2}} dx +C\right)\\ &=e^{-\frac{x^2}{2}}\left( e^{\frac{x^2}{2}}+C\right) \end{align} The general solution is $$y= e^{-\frac{x^2}{2}}\left( e^{\frac{x^2}{2}}+C\right) .$$ Which simplifies to: $$y=Ce^{-\frac{x^2}{2}}+1$$
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