Chapter 6.4, Problem 3E

To determine

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

Answer to Problem 3E

  $y=x$ .

The solution is valid for all the real numbers.

Explanation of Solution

Given:

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

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,

  $\frac{dy}{dx}=\frac{y}{x}$

By separating the variables, it follows that

  $\frac{dy}{y}=\frac{dx}{x}$

Now by integrating on both sides, it follows that

  $\begin{array}{c}{\displaystyle \int \frac{dy}{y}}={\displaystyle \int \frac{dx}{x}}\\ \ln \left|y\right|=\ln \left|x\right|+C_1\\ \ln \left|y\right|-\ln \left|x\right|=C_1\\ \ln \left|\frac{y}{x}\right|=C_1\\ \frac{y}{x}=e^{C_1}\\ \frac{y}{x}=C,\text{where}{e}^{C_1}=C\end{array}$

Now by using the given initial conditions, it follows that

  $\begin{array}{l}\frac{2}{2}=C\\ C=1\end{array}$

Now put $C=1$ in $\frac{y}{x}=C$ , it follows that

  $\begin{array}{l}\frac{y}{x}=1\\ y=x\end{array}$

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