Question

$\frac{dy}{dx}=5y,y\left(0\right)=1$

Short answer

The solution is$y\left(x\right)=e^{5x}$

Step-by-step solution

Step 1. Given information

Given differential equation is$\frac{dy}{dx}=5y,y\left(0\right)=1$

Step 2. Solve the equation 

$$\begin{gathered} \frac{dy}{dx}=5y,y\left(0\right)=1 \\ \int \frac{1}{y}dy=\int 5dx \\ \ln \left(\left|y\right|\right)=5x+c \\ y=e^{5x+c} \\ =A{e}^{5x+c} \\ A=1 \\ y\left(x\right)=e^{5x} \end{gathered}$$

Step 3. The solution 

The solution is$y\left(x\right)=e^{5x}$