Chapter 9, Problem 1RE

In Problems 1 and 2, show that the given function is the general solution of the indicated differential equation.

1.     $y=C\sqrt{x};2x{y}^{\prime }=y$

To determine

To show: The function $y=C\sqrt{x}$ is the general solution of the differential equation $2x{y}^{\prime }=y$.

Explanation of Solution

The differential equation is $2x{y}^{\prime }=y$.

Given that $y=C\sqrt{x}$. Then, $y^{\prime }=\frac{C}{2\sqrt{x}}$.

Substituting $y=C\sqrt{x}$ and $y^{\prime }=\frac{C}{2\sqrt{x}}$ in $2x{y}^{\prime }=y$,

$\begin{array}{l}2x{y}^{\prime }=y\\ 2x\left(\frac{C}{2\sqrt{x}}\right)=C\sqrt{x}\\ C\sqrt{x}=C\sqrt{x}\end{array}$

Therefore, $y=C\sqrt{x}$ is the general solution of the differential equation $2x{y}^{\prime }=y$.