Solved: "what is the differential equation of dy/dx =..."

ichonlarry

Answered question

2022-03-01

what is the differential equation of dy/dx = x e^y

star233

Skilled2022-03-19Added 403 answers

Solve the separable equation $\frac{d y (x)}{d x} = x e^{y (x)}$ :

Divide both sides by $e^{y} (x)$ :
$e^{- y (x)} \frac{d y (x)}{d x} = x$

Integrate both sides with respect to x:
$\int e^{- y (x)} \frac{d y (x)}{d x} d x = \int x d x$

Evaluate the integrals:
$- e^{- y (x)} = \frac{x^{2}}{2} + c_{1}$ , where $c_{1}$ is an arbitrary constant.

Solve for $y (x)$ :
$y (x) = - \log (- \frac{x^{2}}{2} - c_{1})$

Simplify the arbitrary constants:
Answer:
$y (x) = - \log (- \frac{x^{2}}{2} + c_{1})$

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