Question

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

$\frac{dy}{dx}=2xy$

Short answer

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

Step-by-step solution

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

role="math" localid="1649166162207" $\begin{array}{r}\int \frac{1}{y}dy=\int 2xdx\\ \ln \left|y\right|=x^2+C\\ y=e^{x^2+C}\\ =A{e}^{x^2}\left(e^c=A\right)\end{array}$

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