Linear Differential Equation

An $n$th-order ODE is said to be linear in the variable $y$ if $F$ is linear in $\{y,y^{\prime },\dots ,y^{(n)}\}$

Which means the equation can be put in the form:

$$a_n(x)\frac{d^{n}y}{d{x}^n}+a_{n-1}(x)\frac{d^{n-1}y}{d{x}^{n-1}}+\cdots +a_1(x)\frac{dy}{dx}+a_0(x)y=g(x)$$

or, it can be formed into a linear function where the unknowns are of the set $\{y,y^{\prime },y^{\prime \prime },\dots ,y^{(n)}\}$ and the coefficients are functions of $x$.

Where:

• $a_i(x)$ are functions of $x$ only
• $g(x)$ is a function of $x$ only
• $y^{(i)}$ is the $i$th derivative of $y$ with respect to $x$
• $n$ is a non-negative integer representing the order of the differential equation
• $y$ is the dependent variable
• $x$ is the independent variable