# Linear Differential Equations#

## Linear ODEs#

Mathematics Methods 2

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

$\mathcal{L}_x = A_n(x) \frac{d^n}{dx^n} + A_{n-1}(x) \frac{d^{n-1}}{dx^{n-1}} + \cdots + A_1(x) \frac{d}{dx} + A_0(x)$

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 $$1$$:

$\mathcal{L}_x = \frac{d^n}{dx^n} + a_{n-1}(x) \frac{d^{n-1}}{dx^{n-1}} + \cdots + a_1(x) \frac{d}{dx} + a_0(x)$

### 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:

$\mathcal{L}[\alpha u_1 + \beta u_2] = \alpha \mathcal{L}[u_1] + \beta \mathcal{L}[u_2] = 0 + 0 = 0$

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:

$\mathcal{L}[u_p + u_h] = \mathcal{L}[u_p] + \mathcal{L}[u_h] = f + 0 = f$

### 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

## Linear PDEs#

Mathematics for Scientists and Engineers 2