Solve the Differential Equation y’=sqrt(x)/e^y

In this tutorial we shall solve a differential equation of the form $$y’ = \frac{{\sqrt x }}{{{e^y}}}$$ by using the separating the variables method.

The differential equation of the form is given as

\[y’ = \frac{{\sqrt x }}{{{e^y}}}\]

This differential equation can also be written as
\[\frac{{dy}}{{dx}} = \frac{{\sqrt x }}{{{e^y}}}\]

Separating the variables, the given differential equation can be written as
\[{e^y}dy = \sqrt x dx\,\,\,\,{\text{ – – – }}\left( {\text{i}} \right)\]

Keep in mind that in the separating variable technique the terms $$dy$$ and $$dx$$ are placed in the numerator with their respective variables.

Now integrating both sides of the equation (i), we have
\[\begin{gathered} \int {{e^y}dy = \int {\sqrt x dx} } \\ \Rightarrow \int {{e^y}dy = \int {{x^{\frac{1}{2}}}dx} } \\ \end{gathered} \]

Using the formulas of integration $$\int {{e^x}dx = {e^x}} $$ and $$\int {{x^n}dx = \frac{{{x^{n + 1}}}}{{n + 1}}} $$, we get
\[\begin{gathered} {e^y} = \frac{{{x^{\frac{1}{2} + 1}}}}{{\frac{1}{2} + 1}} + c \\ \Rightarrow {e^y} = \frac{{{x^{\frac{3}{2}}}}}{{\frac{3}{2}}} + c \\ \Rightarrow {e^y} = \frac{2}{3}{x^{\frac{3}{2}}} + c \\ \Rightarrow y = \ln \left( {\frac{2}{3}{x^{\frac{3}{2}}} + c} \right) \\ \end{gathered} \]

This is the required solution of the given differential equation.