Separable Equations | Differential Equations

Separable Equations

Sharan Sajiv Menon - October 5th, 2021

A separable equation is a differential equation which can be written in the form $y’=f(x)g(y)$ or $\frac{dy}{dx}=f(x)g(y)$

Examples include:

Separable equations are solved by separation of variables.

Separation of Variables

Check for any values of $y$ that make $g(y)=0$ . Those are the constant solutions
Rewrite the equation so that the y-variables are on the dy side and the x variables are on the dx side
Integrate: $\int \frac{1}{g(y)}dy=\int f(x)dx$
Solve the resulting equation for $y$

Example 1: Solve the following differential equation: $\frac{dy}{dx}=xy$

To rewrite this equation, we can start by multiplying both sides by $dx$ , to remove it from the left side.

dy=xydx

Now, we need to move the y-variable to a different side, to ensure complete separation of variables. We can do this by dividing both sides by $y$ .

\frac{dy}{y}=xdx

Now, we can integrate.

\int \frac{dy}{y}=\int xdx \longrightarrow ln(y)=\frac{x^2}{2} + C

The general solution to a differential equation is in a form of $y=$ , so we need to solve for y.

y=e^{\frac{x^2}{2} + C}= e^{C}e^{\frac{x^2}{2}} = Ce^{\frac{x^2}{2}}

Note here that we can simply rewrite $e^c$ as $C$ , since $e^c$ is a constant. So, our general solution then becomes

y=Ce^{\frac{x^2}{2}}

Example 2: Solve the Differential Equation $y’=(x^2-4)(3y-2)$

Solution: Use separation of variables to solve the ODE.

\frac{dy}{dx}=(x^2-4)(3y-2)\rightarrow \frac{dy}{dx(3y-2)}=x^2-4 \rightarrow \frac{1}{3y-2}dy=(x^2-4)dx

Integrate the separated DE and solve for $y$ :

\int \frac{1}{3y-2}dy = \int (x^2-4)dx \longrightarrow \frac{1}{3}ln|3y-2|=\frac{x^3}{3}-4x +C_1 \longrightarrow ln|3y-2|=x^3-12x + C_1 \longrightarrow \\ 3y-2=e^{x^3-12x+C_1}\rightarrow y=\frac{C_1e^{x^3-12x}+2}{3}

Our final answer is $y=\frac{C_1e^{x^3-12x}+2}{3}$ .

Note that we jumped directly from $e^{x^3-12x+C_1}$ to $C_1e^{x^3-12x}$ . This is because $e^{x^3-12x+C_1}=$