Q. 24 - Calculus

Question

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

      $\frac{dy}{dx}=x{y}^2$

Step-by-Step Solution

Answer

Ans:  The solution of the differential equation $\frac{dy}{dx}=x{y}^2$ is $y=-\frac{2}{x^2+C}$.

Step 1. Given information.

given,

       $\frac{dy}{dx}=x{y}^2$

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.

    $\frac{dy}{dx}=x{y}^2 .....\left(1\right)$

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 $q\left(y\right)=y^2$. So the differential equation can be solved by applying the variable separable method. Separate the variables and integrate both the sides

      $\begin{array}{r}\int \frac{1}{y^2}dy=\int xdx \\ -\frac{1}{y}=\frac{1}{2}{x}^2+C_1 \\ y=-\frac{2}{x^2+2C_1} \\ =-\frac{2}{x^2+C}\left(2C_1=C\right)\end{array}$

Hence a solution to the differential equation $\frac{dy}{dx}=x{y}^2$ is $y=-\frac{2}{x^2+C}$.