Q. 23 - Calculus

Question

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

      $\frac{dy}{dx}=x^{2}y$

Step-by-Step Solution

Answer

Ans:  The solution of the differential equation $\frac{dy}{dx}=x^{2}y$is $y=A{e}^{\frac{1}{3}{x}^3}$

Step 1. Given information.

given,

      $\frac{dy}{dx}=x^{2}y$

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.

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

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

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

    Hence a solution to the differential equation $\frac{dy}{dx}=x^{2}y$ is $y=A{e}^{\frac{1}{3}{x}^3}$