Chapter 6.4, Problem 4E

To determine

To solve: the given initial value problem by using the separation of variables.

Answer to Problem 4E

  $y=3e^{x^2}$ .

The solution is valid for all the real numbers.

Explanation of Solution

Given:

  $\frac{dy}{dx}=2xy$ and $y=3$ when $x=0$ .

Concept used:

A differential equation of the form $\frac{dy}{dx}=f\left(y\right)g\left(x\right)$ is called separable.

The variables can be separated by writing it in the form $\frac{1}{f\left(y\right)}dy=g\left(x\right)dx$ .

The solution is found by anti differentiating each side with respect to its thusly isolated variable.

Calculation:

Consider the given differential equation,

  $\frac{dy}{dx}=2xy$

By separating the variables, it follows that

  $\frac{dy}{y}=2xdx$

Now by integrating on both sides, it follows that

  $\begin{array}{c}{\displaystyle \int \frac{dy}{y}}={\displaystyle \int 2xdx}\\ \ln \left|y\right|=x^2+C\end{array}$

Now by using the given initial conditions, it follows that

  $\begin{array}{c}\ln \left|3\right|=0^2+C\\ C=\ln 3\end{array}$

Now put $C=\ln 3$ in $\ln \left|y\right|=x^2+C$ , it follows that

  $\begin{array}{c}\ln \left|y\right|=x^2+\ln 3\\ \ln \left|y\right|-\ln 3=x^2\\ \ln \left|\frac{y}{3}\right|=x^2\\ \frac{y}{3}=e^{x^2}\\ y=3e^{x^2}\end{array}$

Clearly, the solution is valid for all the real numbers.