Differential Equations and Linear Algebra (4th Edition)

Published by Pearson
ISBN 10: 0-32196-467-5
ISBN 13: 978-0-32196-467-0

Chapter 1 - First-Order Differential Equations - 1.6 First-Order Linear Differential Equations - Problems - Page 59: 23

Answer

$y=c_1\ln x +x^3+c^2$

Work Step by Step

We are given: $\frac{d^2y}{dx^2}+\frac{1}{x}\frac{dy}{dx}=9x$ Let $t=\frac{dy}{dx}$ $\frac{dv}{dx}+\frac{1}{x}v=9x$ The general solution is: $v=I^{-1}(c_1+ \int 9x I dx)$ Intergrating factor: $I=e^{\int \frac{1}{x}dx}=e^{\ln x}=x$ Multiply both sides by the intergrating factor: $\frac{dy}{dx}=x^{-1}(c_1+ \int 9x^2dx)=\frac{1}{x}c_1+3x^2$ Integrate both sides: $y=c_1\ln x +x^3+c^2$
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