Question

Use antidifferentiation and/or separation of variables to solve each of the initial-value problems in Exercises 29–52.

37.$\frac{dy}{dx}=xy,y\left(0\right)=-1$

Short answer

The answer is$y\left(x\right)=-e^{\frac{1}{2}{x}^2}$

Step-by-step solution

Step 1. Given information

We have been given$\frac{dy}{dx}=xy,y\left(0\right)=-1$

Step 2. Solve using antidifferentiation and/or variable separable method.

The differential equation can be solved by applying variable separable method.

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

Now put $y\left(0\right)=-1$,

$$\begin{gathered} -1=A{e}^{\frac{1}{2}{\left(0\right)}^2} \\ -1=A \end{gathered}$$

So, the answer is$y\left(x\right)=-e^{\frac{1}{2}{x}^2}$