Threads

Write a linear equation as

If $y_p$ is one particular solution and $y_h$ is any homogeneous solution, then the complete solution is

For suitable forcing functions:

• a polynomial of degree $n$ suggests a general polynomial of degree $n$;
• $e^{\beta x}$ suggests $Ce^{\beta x}$;
• $p_n(x)e^{\beta x}$ suggests $q_n(x)e^{\beta x}$, where $q_n$ has degree $n$;
• $\sin(\beta x)$ or $\cos(\beta x)$ suggests $$C_1\cos(\beta x)+C_2\sin(\beta x);$$
• $e^{\alpha x}\sin(\beta x)$ or $e^{\alpha x}\cos(\beta x)$ suggests $$e^{\alpha x} \bigl(C_1\cos(\beta x)+C_2\sin(\beta x)\bigr).$$

Substitute the trial into the equation and match coefficients. If the trial overlaps with a homogeneous mode whose characteristic root has multiplicity $s$, multiply the whole trial by $x^s$ (so a simple overlap requires one factor of $x$, while a repeated root requires $x^2$).

The kernel of a linear differential operator $T$ is

If $T[y]=f$ and $y_p$ is one particular solution, then

Equivalently, every solution is $y_p+y_h$ for some homogeneous solution $y_h$.

If $T:V\to W$ is a linear map, its kernel is

The same set is also called the null space. If $A\in\mathbb F^{m\times n}$ is regarded as the map $\mathbb F^n\to\mathbb F^m$, then

Let $A$ be a matrix over an infinite scalar field $\mathbb F$ (such as $\mathbb R$ or $\mathbb C$), let $b$ be a vector of the appropriate dimension and let $[A\mid b]$ be the augmented matrix formed by appending $b$ to $A$. The equation $Ax=b$ has:

1. no solution if $$\operatorname{rank}(A)\ne\operatorname{rank}([A\mid b]);$$
2. a unique solution if $$\operatorname{rank}(A)=\operatorname{rank}([A\mid b]) \quad\text{and}\quad \operatorname{nullity}(A)=0;$$
3. infinitely many solutions if $$\operatorname{rank}(A)=\operatorname{rank}([A\mid b]) \quad\text{and}\quad \operatorname{nullity}(A)>0.$$

In the last case, if $\operatorname{nullity}(A)=k$, then every solution has the form

where $x_p$ is any particular solution of $Ax=b$, $\{u_1,\ldots,u_k\}$ is a basis for $\ker(A)$ and $\omega_1,\ldots,\omega_k\in\mathbb F$.