# Linear Differential Equations

## Contents

# Linear Differential Equations#

## Linear ODEs#

**Definition.** An ODE is **linear** if it is of the form \(\mathcal{L}_x[u] = f\), where \(\mathcal{L}_x\) is the linear differential operator

Linear ODEs of the form \(\mathcal{L}[u] = 0\) are termed **homogeneous**, otherwise **inhomogeneous**.

We will often want to work with the linear operator in *standard form*, meaning that the factor multiplying the highest derivative is \(1\):

### Solutions#

Solutions of a homogeneous ODE satisfy the **superposition principle** (or *linearity principle*): if \(u_1\) and \(u_2\) are solutions of a linear ODE \(\mathcal{L}[u] = 0\) then any linear combination \(\alpha u_1 + \beta u_2\) is also a solution:

A **general solution** of an ODE is a solution \(u = \alpha u_1 + \beta u_2\) where \(u_1\) and \(u_2\) are linearly independent and they are called a **basis** of the set of solutions. A **particular solution** is obtained by assigning specific values to constants.

Every element of the general solution of an ODE is a function of the form \(u = u_p + u_h\), where \(u_h\) is an element of the general solution of the homogeneous ODE and \(u_p\) is a **particular integral**, the solution of the inhomogeneous problem:

### Solution methods#

**Separation of variables**when we can algebraically transform the equation to the form \( f(u) du = g(x) dx \) which we can integrate directly**Variation of parameters**where we find the solution to the homogeneous problem and use it as solution*ansatz*for a particular solution by allowing constants to become new unknown functions which we then seek**Undetermined coefficients****Green’s functions****Numerical methods**