Q. 22 - Calculus

Question

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

        $\frac{dy}{dx}=-3xy$

Step-by-Step Solution

Answer

Ans:  The solution of the differential equation  $\frac{dy}{dx}=-3xy$is $y=A{e}^{-\frac{3}{2}{x}^2}$.

Step 1. Given information.

given,

       $\frac{dy}{dx}=-3xy$

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.

    $\frac{dy}{dx}=-3xy .....\left(1\right)$

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 $p\left(x\right)=-3x$ and $q\left(y\right)=y$. So the differential equation can be solved by applying variable separable method. Separate the variables and integrate both the sides

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

Hence a solution to the differential equation $\frac{dy}{dx}=-3xy$ is $y=A{e}^{-\frac{3}{2}{x}^2}$