Question

Use antidifferentiation and/or separation of variables to solve each of the differential equations in Exercises 19–28. Your answers will involve unsolved constants.

28.$\frac{dy}{dx}=x^2{e}^{-y}$

Short answer

The solution of the differential equation is$y\left(x\right)=\ln \left|\frac{1}{3}{x}^3+C\right|$

Step-by-step solution

Step 1. Given information

We have been given the differential equation as$\frac{dy}{dx}=x^2{e}^{-y}$

Step 2. Solve using antidifferentiation and/or variable separable method.

The differential equation can be solved by applying variable separable method.

$$\begin{gathered} \frac{dy}{dx}=x^2{e}^{-y} \\ \int \frac{1}{e^{-y}}dy=\int x^{2}dx \\ \int e^{y}dy=\frac{1}{3}{x}^3+C \\ {e}^y=\frac{1}{3}{x}^3+C \\ y=\ln \left|\frac{1}{3}{x}^3+C\right| \end{gathered}$$