What are Differential Equations?

A differential equation is simply a equation with a derivative in it. The following equation is considered a differential equation.

This equation can also be considered a differential equation.

Both these equations have derivatives in them. The purpose of this guide is to serve as a compilation of my notes for differential equations. I include many solved problems to help understand the concepts presented.

First Order Differential Equations

First order differential equations are equations that have only the first derivative in it ($\frac{dy}{dx})$.

Example: Solve the First Order DE $\frac{dy}{dx}=2x$.

Solution: This is a very simple differential equation and it only requires a simple integration. To solve for $y$, we simply move the $dx$ over to the right side and integrate both sides.

Solving this integral will gives us the following equation.

This is known as the general solution of the Differential Equation. There is another type of solution known as a particular solution, which will be covered in the Initial Value Problem Section (IVP).

We used a technique known as Separation of Variables, which will be discussed in more detail in a later section.

Linear First Order DE's

A first order differential equations is linear if it can be written as:

If $f(x)=0$, then the DE is homogenous, otherwise it is non-homogenous. A first order DE that cannot be written in the above form is non-linear.

Example: Find the general solution $y'-ay=0$​

1. Rewrite: $\frac{y'}{y}=a$
2. Integrate: $\int \frac{1}{y}dy=\int adx \rightarrow ln|y|=ax+k$
3. Isolate $y$: $y=e^ke^{ax}$
4. Rewrite $e^k$ as $C$: $y=ce^{ax}$.

You can also rewrite the equation to $\frac{dy}{dx}=ay$, and use separation of variables to solve it.