 # What is a solution to the differential equation dy/dx=x+y? Joyce Sharp 2022-09-18 Answered
What is a solution to the differential equation $\frac{dy}{dx}=x+y$?
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Let u=x+y
$⇒\frac{du}{dx}=\frac{d}{dx}\left(x+y\right)=1+\frac{dy}{dx}$
$⇒\frac{dy}{dx}=\frac{du}{dx}-1$
Thus, making the substitutions into our original equation,
$\frac{du}{dx}-1=u$
$⇒\frac{du}{u+1}=dx$
$⇒\int \frac{du}{u+1}=\int dx$
$⇒\mathrm{ln}\left(u+1\right)=x+{C}_{0}$
$⇒{e}^{\mathrm{ln}\left(u+1\right)}={e}^{x+{C}_{0}}$
$⇒u+1={C}_{1}{e}^{x}\phantom{\rule{1ex}{0ex}}\text{}\phantom{\rule{1ex}{0ex}}$ (where ${C}_{1}={e}^{{C}_{0}}$)
Substituting x+y=u back in,
$⇒x+y+1={C}_{1}{e}^{x}$
$\therefore y={C}_{1}{e}^{x}-x-1$

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