Question

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

30.$\frac{dy}{dx}=3x,y\left(0\right)=4$

Short answer

The answer is$y\left(x\right)=\frac{3}{2}{x}^2+4$

Step-by-step solution

Step 1. Given information

We have been given$\frac{dy}{dx}=3x,y\left(0\right)=4$

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

The differential equation can be solved by antidifferentiating.

$$\begin{gathered} \int dy=\int 3xdx \\ y=\frac{3}{2}{x}^2+C \end{gathered}$$

Now put $y\left(0\right)=4$,

$$\begin{gathered} 4=\frac{3}{2}{\left(0\right)}^2+C \\ 4=C \end{gathered}$$

So, the answer is$y\left(x\right)=\frac{3}{2}{x}^2+4$