Differential Equations

A differential equation relates a function to its derivatives with respect to independent variables. The general form of a linear ODE is

$$a_n(x)\,y^{(n)} + a_{n-1}(x)\,y^{(n-1)} + \dots + a_2(x)\,y'' + a_1(x)\,y' + a_0(x)\,y = f(x)$$

Order – the degree of the highest derivative
Ordinary – a function with only one variable
Partial – a function under multiple variables
ODE – “Ordinary Differential Equation”
Linear – every derivative appears to the first power
Homogeneous – $f(x) = 0$; every term involves $y$ or its derivatives

First Order ODE

An Ordinary Differential Equation is an equation with derivatives taken with respect to one variable.

The Picard-Lindelöf Theorem guarantees that a unique solution exists for an initial value problem as long as the function and its partial derivative with respect to $y$ are continuous in a neighborhood of the initial condition.

Linear ODE

A first order linear ODE generally follows the structure

$$y' + P(x)\,y = Q(x)$$

where $y$ and its derivatives each appear to the first power.

An algebraic linear equation: $ay_1 + y_2 = c$

A differential linear equation: $a(x)\,y' + b(x)\,y = c(x)$

Therefore $y' + x^2 y = 5$ is linear while $y' + y^2 = 0$ is not linear.

Homogeneity

A linear ODE is homogeneous when $f(x) = 0$ — that is, every term on the left-hand side involves $y$ or one of its derivatives, and there is no standalone function of $x$.

The following is homogeneous: $y'' + 2y = 0$

The following is non-homogeneous because of the constant term $-5$: $y''' + 4x^7 y'' + \sin(x)\,y - 5 = 0$

Separable ODE

Whenever a Linear First Order Differential Equation is Homogeneous, the equation can be separated. An ODE expressed in the form

$$\frac{dy}{dx} = g(x)\,y$$

can be decomposed into

$$\frac{1}{h(y)}\,dy = g(x)\,dx \qquad \text{or} \qquad \int \frac{1}{h(y)}\,dy = \int g(x)\,dx$$

which reduces to

$$\ln(y) = H(x) + C \qquad \text{or} \qquad y = C\,e^{H(x)}$$

Example:

$$\frac{dy}{dx} = 4y\sin(3x) \tag{1}$$ $$\frac{1}{y}\,dy = 4\sin(3x)\,dx \tag{2}$$ $$\int \frac{1}{y}\,dy = \int 4\sin(3x)\,dx \tag{3}$$ $$\ln(y) = -\frac{4}{3}\cos(3x) + C \tag{4}$$ $$y = C\,e^{-\frac{4}{3}\cos(3x)} \tag{5}$$

The ability to separate variables and integrate both sides independently is what makes these first-order homogeneous linear ODEs the most approachable starting point. From here, techniques like integrating factors and variation of parameters extend the same ideas to non-homogeneous and higher-order equations.