Question

Verify that $y\left(x\right)=\sqrt{x+9}$is a solution of the initial-value problem $\frac{dy}{dx}=\frac{1}{2y}$and $y\left(0\right)=3$.

Short answer

The solution of the given differential equation has been verified.

Step-by-step solution

Step 1. Given information:

The differential equation:$\frac{dy}{dx}=\frac{1}{2y}$

$y\left(0\right)=3$

The solution of this differential equation$y\left(x\right)=\sqrt{x+9}$

Step  2. Verify the solution.

$$\begin{gathered} \frac{dy}{dx}=\frac{1}{2y} \\ 2ydy=dx \end{gathered}$$

Now integrate both sides.

$$\begin{gathered} \int 2ydy=\int dx \\ 2\int ydy=\int dx \\ 2.\frac{y^2}{2}=x+c \\ {y}^2=x+c \\ y=\sqrt{x+c} \end{gathered}$$

Now find $c$by substituting $y\left(0\right)=3$

$$\begin{gathered} 3=\sqrt{0+c} \\ 3=\sqrt{c} \\ c=9 \end{gathered}$$

So, the solution is:$y\left(x\right)=\sqrt{x+9}$

Thus, the solution has been verified.