Question

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

34.$\frac{dy}{dx}=\frac{1}{y},y\left(0\right)=3$

Short answer

The answer is $y\left(x\right)=\sqrt{2x+9}$

Step-by-step solution

Step 1. Given information

We have been given$\frac{dy}{dx}=\frac{1}{y},y\left(0\right)=3$

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

The differential equation can be solved by antidifferentiating.

$$\begin{gathered} \int ydy=\int dx \\ \frac{1}{2}{y}^2=x+C_1 \\ {y}^2=2x+C(2C_1=C) \end{gathered}$$

Now put $y\left(0\right)=3$,

$$\begin{gathered} {\left(3\right)}^2=2\left(0\right)+C \\ 9=C \end{gathered}$$

So,

$$\begin{gathered} y^2=2x+9 \\ y=\sqrt{2x+9} \end{gathered}$$