Modification if trial form is in $y_g(x)$ $A e^{\alpha x}$ $C e^{\alpha x}$ Multiply by $x$ (if $\alpha$ is a simple root of aux. eq.) or $x^2$ (if $\alpha$ is a double root). $A \cos(\beta x) + B \sin(\beta x)$ $C \cos(\beta x) + D \sin(\beta x)$ Multiply by $x$ (if $\pm i\beta$ are roots of aux. eq.) Polynomial $P_n(x)$ of degree $n$ $Q_n(x)$ (general polynomial of degree $n$) Multiply by $x$ (if $0$ is a root of aux. eq.) or $x^2$ (if $0$ is a double root).

Substitute $y_p(x)$ and its derivatives into the nonhomogeneous ODE.

Equate coefficients to find the undetermined constants.

Method of Variation of Parameters (for $y'' + P(x)y' + Q(x)y = R(x)$):

Find two linearly independent solutions $y_1(x), y_2(x)$ of the homogeneous equation $y'' + P(x)y' + Q(x)y = 0$.
Calculate the Wronskian $W(y_1, y_2)(x) = y_1 y_2' - y_2 y_1'$.
The particular solution is $y_p(x) = -y_1(x) \int \frac{y_2(x)R(x)}{W(y_1, y_2)(x)}dx + y_2(x) \int \frac{y_1(x)R(x)}{W(y_1, y_2)(x)}dx$.

Important Theorems

Existence & Uniqueness (Second Order Linear): If $P(x), Q(x), R(x)$ are continuous on $[a,b]$, then for any $x_0 \in [a,b]$ and $y_0, y_0'$ any numbers, $y'' + P(x)y' + Q(x)y = R(x)$ has a unique solution $y(x)$ on $[a,b]$ satisfying $y(x_0)=y_0$ and $y'(x_0)=y_0'$.
Principle of Superposition (Homogeneous): If $y_1, y_2$ are solutions of a homogeneous linear ODE, then $c_1 y_1 + c_2 y_2$ is also a solution.
Wronskian Test for Linear Independence: Two solutions $y_1, y_2$ of $y'' + P(x)y' + Q(x)y = 0$ are linearly independent iff $W(y_1, y_2)(x) \neq 0$ for all $x$ in the interval. If $W(y_1, y_2)(x_0) = 0$ for some $x_0$, then $W(y_1, y_2)(x) = 0$ for all $x$.
Abel's Formula: For solutions $y_1, y_2$ of $y'' + P(x)y' + Q(x)y = 0$, $W(y_1, y_2)(x) = C e^{-\int P(x)dx}$.

Watch Out!

For undetermined coefficients, remember to modify the trial solution if it overlaps with the homogeneous solution.
Variation of parameters is more general (works for variable coefficients) but often involves harder integrals than undetermined coefficients.
Ensure the ODE is in standard form ($y'' + P(x)y' + Q(x)y = R(x)$) before applying Variation of Parameters.

Module 3: Systems of ODEs

Key Definitions

System of First-Order ODEs: A set of $n$ equations where the derivatives of $n$ unknown functions are expressed in terms of the independent variable and the functions themselves.
Homogeneous Linear System: A system of linear ODEs where all constant terms are zero.
Nonhomogeneous Linear System: A system of linear ODEs with at least one non-zero constant term.
Trivial Solution (for systems): A solution where all unknown functions are identically zero.
Wronskian (for systems): For two solutions $\begin{pmatrix} x_1(t) \\ y_1(t) \end{pmatrix}$ and $\begin{pmatrix} x_2(t) \\ y_2(t) \end{pmatrix}$ of a 2x2 system, $W(t) = x_1(t)y_2(t) - x_2(t)y_1(t)$.
Linearly Dependent Solutions (for systems): Two solutions are linearly dependent if one is a constant multiple of the other.
Auxiliary Equation (for constant coeff