Lesson 5 - Separable Equations

In some situations, we can separate an equation by it's variables through products or division. These equations are called separable differential equations.

Separable differential equations are characterized by the form $\frac{dy}{dx}=g(x)p(y)$. Notice we can use division to separate the variables by $x$ and $y$.

Classify as separable or not separable DEs.
a) $y^{\prime }=x^2{e}^y\cos y$
b) $\frac{dy}{dx}=x^{2}y-1$
c) $y^{\prime }=e^{xy}\cos y$
d) $y^{\prime }=(x^2+y^2)y^3$

a) We can easily separate $x^2$ from $e^y\cos y$ by division. This is a separable DE.
b) Notice that we cannot separate $x^2$ from $y$ because we are subtracting $1$. If we add $1$ to each side, this would create the same issue for separating $\frac{dy}{dx}$. This is not a separable DE.
c) We have an exponent consisting of $x$ and $y$. $e^{xy}$ can be separated through the laws of exponents, $e^x+e^y$. However, since these exponents are added, we are cannot separate the variables.
d) Because $x^2$ and $y^2$ are added, we cannot separate the DE.

Note: Some DEs may appear inseparable due to adding or subtracting. However, recall that factoring is a tool that we can use to separate variables that are added.

Example:

$y^{\prime }=(x^2{y}^2+2y^2)$

$y^{\prime }=y^2(x^2+2)$

This is now a separable DE.

Let's solve this differential equation.

Left side, using power rule:

Right side, similarly using power rule:

Now, we solve for y.

You can choose to further simplify to provide a cleaner solution.

Differential equations often require various techniques of manipulation to be executed. When we are tasked with finding solutions to differential equations that truly are inseparable, there are a few tools we can use to accomplish this. Let's move on to one of the first tools we can use in solving first order differential equations.

Next Lesson: Lesson 6 - Linear First-Order Equations

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