Q. 27 - Calculus

Question

Use antidifferentiation and/or separation of variables to solve the given differential equations. Your answers will involve unsolved constants. 

     $\frac{dy}{dx}=x{e}^{-y}$

Step-by-Step Solution

Answer

Ans:  The solution of the differential equation  $\frac{dy}{dx}=x{e}^{-y}$ is $y=\ln \left|\frac{1}{2}{x}^2+C\right|$

Step 1. Given information.

given,

     $\frac{dy}{dx}=x{e}^{-y}$

Step 2. Consider the differential equation defined by equation (1) given below and solve it by using antidifferentiation and/or separation of the variable method.

   $\frac{dy}{dx}=x{e}^{-y} ....\left(1\right)$ 

Step 3. Solution

Note that the differential equation (1) is of the form of $\frac{dy}{dx}=p\left(x\right)q\left(y\right)$ in which $p\left(x\right)=x$ and $q\left(y\right)=e^{-y}$. So the differential equation can be solved by applying the variable separable method. Separate the variables and integrate both the sides

      $\begin{array}{r}\int \frac{1}{e^{-y}}dy=\int xdx\\ \int e^{y}dy=\frac{1}{2}{x}^2+C\\ e^y=\frac{1}{2}{x}^2+C\\ y=\ln \left|\frac{1}{2}{x}^2+C\right|\end{array}$

Hence a solution to the differential equation $\frac{dy}{dx}=x{e}^{-y}$ is $y=\ln \left|\frac{1}{2}{x}^2+C\right|$