Question

In Problems 1–22 Solve the given differential equation by separation of variables.$\frac{\mathbf{\text{dS}}}{\mathbf{\text{dr}}}\mathbf{\text{=ks}}$

Short answer

$S=C{e}^{kr}$

Step-by-step solution

Step 1:Definition of separable equation

A first-order differential equation of the form$\frac{dy}{dx}=g\left(x\right)h\left(y\right)$is said to be separable or to have separable variables.

Step 2:Separate the variables and integration

Separate the variable

$\frac{dS}{S}=kdr$

Integrate both sides

$\int \frac{dS}{S}=\int kdr$

Do the integration

$\ln S=kr+c_1$

Relabeling$e^{c_1}=C$

$\begin{array}{c}S=e^{kr+c_1}\\ S=C{e}^{kr}\end{array}$

Hence the solution is${\mathbf{\text{S=Ce}}}^{\mathbf{\text{kr}}}$