Question

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

32.$\frac{dy}{dx}=-5y,y\left(0\right)=2$

Short answer

The answer is$y\left(x\right)=2e^{-5x}$

Step-by-step solution

Step 1. Given information

We have been given$\frac{dy}{dx}=-5y,y\left(0\right)=2$

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

The differential equation can be solved by antidifferentiating.

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

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

$$\begin{gathered} 2=A{e}^{-5\left(0\right)} \\ 2=A \end{gathered}$$

So, the answer is$y\left(x\right)=2e^{-5x}$