Linear Differential Equations

Linear ODEs

Linear Operator

An operator \(L\) is linear if:

\[ L(f + g) = Lf + Lg \]

\[ L(\lambda f) = \lambda Lf \]

for functions \(f, g\) and a scalar \(\lambda\).

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