How to solve the differential equation: $y-xy'=\exp(y')$

Question

$$y-xy'=\exp(y')$$

I want to solve this differential equation, which looks simple but hard to solve. Any method?

Answer 1

Use Clairaut equation to find the solution.

The general solution is $$y(x)=Cx+e^{C}.$$

Proof from Wikipedia article.

Differentiate both sides with respect to $x$ $$y'=y'+xy''+\exp\left(y'\right)y'',$$ so $$\left[x+\exp\left(y'\right)\right]y'' = 0.$$

General solution: $$y'' = 0 \Longrightarrow y(x)=Cx+e^C.$$

Singular solution: $$x+\exp\left(y'\right) = 0.$$

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Thank you @zwim: Desmos calculator.