Question

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

36.$\frac{dy}{dx}=\frac{6}{x},y\left(1\right)=7$

Short answer

The answer is$y\left(x\right)=6\ln \left|x\right|+7$

Step-by-step solution

Step 1. Given information

We have been given$\frac{dy}{dx}=\frac{6}{x},y\left(1\right)=7$

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

The differential equation can be solved by antidifferentiating.

$$\begin{gathered} \int dy=\int \frac{6}{x}dx \\ y=6\ln \left|x\right|+C \end{gathered}$$

Now put $y\left(1\right)=7$,

$$\begin{gathered} 7=6\ln 1+C \\ 7=C \end{gathered}$$

So, the answer is$y\left(x\right)=6\ln \left|x\right|+7$