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 - 1.3 The Geometry of First-Order Differential Equations - Problems: 10

Answer

$y=x^3$, See below

Work Step by Step

Find $y'$ by taking the derivative in respect to $x$ of the equation. $$y=cx^3$$ $$y'=3cx^2$$ Substitute $y'=3cx^2$ and $y=cx^3$ into the differential equation to get $$y'=3y/x$$ $$3cx^2=3cx^3/x=3cx^2$$ This is a true statement, therefore, this is a solution to the differential equation. Find the value of $y$ at $x=2$. $$y(2)=c(2)^3=8c$$ Since $y(2)=8$, the equation becomes $$y(2)=8=8c$$ $$c=1$$ Therefore, the equation is equal to $y=x^3$.
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