Question

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

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

Short answer

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

Step-by-step solution

Step 1. Given information

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

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

The differential equation can be solved by antidifferentiating.

$$\begin{gathered} \int \frac{1}{y}dy=\int 3dx \\ \ln \left|y\right|=3x+C \\ y=e^{3x+C} \\ y=A{e}^{3x}\left(e^c=A\right) \end{gathered}$$

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

$$\begin{gathered} 4=A{e}^{3\left(0\right)} \\ 4=A \end{gathered}$$

So, the answer is$y=4e^{3x}$