15. \frac{dS}{dr}=kS solve by separable method

\frac{dS}{dr}=kS solve by separable method.

To solve the differential equation \frac{dS}{dr}=kS by separable method, we need to separate the variables S and r on opposite sides of the equation and integrate both sides.

Starting with the given differential equation:

    \begin{equation*} \frac{dS}{dr}=kS \end{equation*}

We can divide both sides by S to obtain:

    \begin{equation*} \frac{1}{S}dS=kdr \end{equation*}

Now we can integrate both sides with respect to their respective variables:

    \begin{align*} \int \frac{1}{S}dS &= \int kdr \end{align*}

    \begin{align*}ln|S| &= kr + C_1 \end{align*}

    \begin{align*}S &= e^{kr+C_1} = Ae^{kr} \end{align*}

where C_1 and A are constants of integration.

Therefore, the general solution to the given differential equation is:

    \begin{equation*} S = Ae^{kr} \end{equation*}

where A is a constant determined by the values of the constants of integration.

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