The Method of Separation of Variables
for First Order Differential Equations.

Use the method of separation of variables to help simplify differential equations of the form:

Assuming $g\left(y\right)\ne 0$, then

$$\frac{1}{g\left(y\right)}\frac{dy}{dx}=f(x)$$

Integrate with respect to $x$

$$\int \frac{1}{g\left(y\right)}\frac{dy}{dx}dx=\int f(x)dx+C$$

where $C$ is a constant.

Using the chain rule on the left expression ...

$$\int \frac{1}{g\left(y\right)}dy=\int f\left(x\right)dx+C$$

Solve each integral, perhaps using the integration flow chart.

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Questions:

Differential Equations [ MS ]
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Differential equations - first order
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1. • Find the particular solution to $$\frac{dy}{dx}=ky$$ if $y=y_0$ when $x=0$
     
     [ Solution ]
   • $$\frac{dy}{dx}=ky$$ $$\frac{1}{y}\frac{dy}{dx}=k$$ $$\int \frac{1}{y}\frac{dy}{dx}dx=k\int dx$$ $$\int \frac{1}{y}dy=kx+C$$ where $C$ is a constant of integration $$lny=kx+C$$ $$y={\mathrm{e}}^{kx+C}$$ $$y={\mathrm{e}}^C{\mathrm{e}}^{kx}$$ Using the boundary condition $y=y_0$ when $x=0$ $$y_0={\mathrm{e}}^C{\mathrm{e}}^{k\times 0}$$ $$y_0={\mathrm{e}}^C$$ So ... $$y=y_0{\mathrm{e}}^{kx}$$
     
     This is useful in physics in areas such as capacitance, nuclear decay, and radiation absorption

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If $g\left(y\right)=0$?

If $g\left(y\right)=0$ for the all $x$, then

$$\frac{dy}{dx}=0$$

and

$$y=\int 0dx+C$$ $$y=C$$

If $g\left(y\right)=0$ for only some of $y$, then the solution needs to be investigated on a case by case bases and is likely to discontinuous.