Answer
$c=-\frac{1}{2}$.
Work Step by Step
Since $y=x-2+e^{cx}$, then we have $y'=1+ce^{cx}$. Now, by substitution, we get
$$2y'+y=2+2ce^{cx}+x-2+e^{cx}=x$$
which leads to
$$(2c+1)e^{cx} =0$$
that is, $2c+1=0$ and hence $c=-\frac{1}{2}$.
Calculus (3rd Edition)
by Rogawski, Jon; Adams, Colin
Published by
W. H. Freeman
ISBN 10:
1464125260
ISBN 13:
978-1-46412-526-3
Chapter 10 - Introduction to Differential Equations - Chapter Review Exercises - Page 525: 2
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