Solve Differential Equation dy/dx=xe^-y

In this tutorial we shall evaluate the simple differential equation of the form \frac{{dy}}{{dx}} = x{e^{ - y}}, and we shall use the method of separating the variables.

The differential equation of the form is given as

\frac{{dy}}{{dx}} = x{e^{ - y}}

Separating the variables, the given differential equation can be written as

\begin{gathered} \frac{1}{{{e^{ - y}}}}dy = xdx \\ \Rightarrow {e^y}dy = xdx\,\,\,\,{\text{ - - - }}\left( {\text{i}} \right) \\ \end{gathered}

With the separating the variable technique we must keep the terms dy and dx in the numerators with their respective functions.

Now integrating both sides of the equation (i), we have

\int {{e^y}dy = \int {xdx} }

Using the formulas of integration \int {{e^x}dx = {e^x}} and \int {{x^n}dx = \frac{{{x^{n + 1}}}}{{n + 1}}} , we get

\begin{gathered} {e^y} = \frac{1}{2}{x^2} + c \\ \Rightarrow y = \ln \left( {\frac{1}{2}{x^2} + c} \right) \\ \end{gathered}

This is the required solution of the given differential equation.