Question

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

35.$\frac{dy}{dx}=\frac{3}{y},y\left(0\right)=2$

Short answer

The answer is$y\left(x\right)=\sqrt{6x+4}$

Step-by-step solution

Step 1. Given information

We have been given$\frac{dy}{dx}=\frac{3}{y},y\left(0\right)=2$

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

The differential equation can be solved by antidifferentiating.

$$\begin{gathered} \int ydy=\int 3dx \\ \frac{1}{2}{y}^2=3x+C_1 \\ {y}^2=6x+2C_1 \\ {y}^2=6x+C \end{gathered}$$

Now put $y\left(0\right)=2$,

role="math" localid="1649147751077" $$\begin{gathered} {\left(2\right)}^2=6\left(0\right)+C \\ 4=C \end{gathered}$$

So,

role="math" localid="1649147767750" $$\begin{gathered} y^2=6x+4 \\ y=\sqrt{6x+4} \end{gathered}$$