Q. 21 - Calculus

Question

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

     

       $\frac{dy}{dx}=2xy$

Step-by-Step Solution

Answer

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

Step 1. Given information.

given,

      $\frac{dy}{dx}=2xy$

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.

    $\frac{dy}{dx}=2xy ....\left(1\right)$

Step 3. Now,

Noe that the differential equation $\left(1\right)$ is of the form of $\frac{dy}{dx}=p\left(x\right)q\left(y\right)$ in which $p\left(x\right)=2x$ and $q\left(y\right)=y$. So the differentialequation can solved by applying variable separable method. Separate the variables and integrate both the sides

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

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