Question

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

$\frac{dy}{dx}=x{y}^2$

Short answer

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

Step-by-step solution

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 $y=-\frac{2}{x^2+C}$.