Ordinary differential equations will not be covered in this chapter as they are an earlier topic. However, I will do a brief definition at the start. PDEs will be the primary focus in this chapter.
Ordinary Differential Equations (ODEs) are differential equations that have a single variable. ODEs have a general form:
$$F\left(x,y,\frac{dy}{dx},\frac{d^{2}y}{dx^{2}},...,\frac{d^{n}y}{dx^{n}}\right)=0$$
where:
Partial Differential Equations (PDEs) are differential equations that have multiple independent variables. Instead of using the standard d, they use partial derivatives (∂) to show the change with respect to multiple variables. The general form is:
$$F\left(x,y,u,\frac{\partial u}{\partial x},\frac{\partial u}{\partial y},\frac{\partial^{2}u}{\partial x^{2}},....\right)=0$$
where:
Fundamentally, imagine an ODE as a means to track a single car, while a PDE tracks all the traffic in the city.
ODEs are usually classified by two primary aspects: their order, i.e., the degree of their derivative, and whether they are linear or non-linear.
First-order ODEs involve only the first derivative. Here is a basic first-order ODE:
$$\frac{dy}{dx}+y=x$$
Second-order ODEs involve up to the second derivative. Here is an example:
$$\frac{d^{2}y}{dx^{2}}+2\frac{dy}{dx}+y=0$$
Higher-order ODEs involve everything from the third derivative or higher. It is unlikely to ever appear in a first-year analysis exam, but you never know.
Linear and Non-linear ODEs: An ODE is linear if the dependent variable and its derivatives appear in a linear form, meaning they are not multiplied together. Anything else is considered non-linear. A linear ODE can be written in the form:
$$a_{n}(x)\frac{d^{n}y}{dx^{n}}+a_{n-1}(x)\frac{d^{n-1}y}{dx^{n-1}}+...+a_{1}(x)\frac{dy}{dx}+a_{0}(x)y=f(x)$$
where:
Here are some basic examples. We will go into much more detail when solving ODEs.
$$\frac{dy}{dx}+3y=x$$, and $$\frac{d^{2}y}{dx^{2}}+x\frac{dy}{dx}+y=\sin x$$
(NON-LINEAR) To be added
A separable ODE can be written as:
$$\frac{dy}{dx}=f(x)g(y)$$
$$\frac{dy}{g(y)}=f(x)dx$$
$$\int\frac{dy}{g(y)}=\int f(x)dx$$
Factor Method
$$\frac{dy}{dx}+P(x)y=Q(x)$$
$$\mu(x)=e^{\int P(x)dx}$$
$$\frac{d}{dx}(\mu(x)y)=\mu(x)Q(x)$$
$$y=\frac{1}{\mu(x)}\int \mu(x)Q(x)dx+C$$
Homogeneous Equations with Constant Coefficients
Not
Nonhomogeneous Equations: Method of Undetermined Coefficients
Finished