Study Notes: Differential Equations and Linear ODEs (Calculus II/III)

Sections 2.1–2.4: Linear Independence and Homogeneous ODEs

Linear Independence and the Wronskian

Understanding whether a set of functions is linearly independent is crucial for solving differential equations. The Wronskian is a determinant used to test for linear independence.

• Wronskian Test: For functions $f_1(x), f_2(x), ..., f_n(x)$, the Wronskian is $W(f_1, f_2, ..., f_n)(x) = \det \begin{pmatrix} f_1(x) & f_2(x) & \cdots & f_n(x) \\ f_1'(x) & f_2'(x) & \cdots & f_n'(x) \\ \vdots & \vdots & & \vdots \\ f_1^{(n-1)}(x) & f_2^{(n-1)}(x) & \cdots & f_n^{(n-1)}(x) \end{pmatrix}$

• If $W \neq 0$ on an interval, the functions are linearly independent there.

Homogeneous Linear ODEs

• Homogeneous ODE: An equation of the form $a_n y^{(n)} + a_{n-1} y^{(n-1)} + \cdots + a_0 y = 0$.

• Constant Coefficient ODEs: Coefficients $a_i$ are constants.

• Characteristic polynomial method is used to solve these equations.

• Handle cases of real, complex, and repeated roots.

• Find roots by factoring, quadratic formula, or polynomial division.

• Euler's Formula: $e^{i\theta} = \cos \theta + i \sin \theta$ is used for complex roots.

• DeMoivre's Theorem: $(\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta)$.

Example

• Given $y'' - 3y' + 2y = 0$, the characteristic equation is $r^2 - 3r + 2 = 0$.

• Roots: $r = 1, 2$; General solution: $y(x) = C_1 e^{x} + C_2 e^{2x}$.

Section 2.5: Initial Conditions and Model Equations

Initial Value Problems (IVP)

Initial conditions specify the value of the solution and its derivatives at a starting point, allowing for a unique solution to an ODE.

• IVP Example: $y'' + 4y = 0$, $y(0) = 2$, $y'(0) = 0$.

• Model equation: $mx'' + cx' + kx = 0$ describes systems like mass-spring-damper.

• Parameters $m$, $c$, $k$ represent mass, damping, and stiffness, respectively.

• Solutions can be overdamped, critically damped, or underdamped depending on the discriminant $c^2 - 4mk$.

Example

• For $m=1$, $c=2$, $k=1$: $x'' + 2x' + x = 0$ is critically damped since $c^2 = 4mk$.

Sections 2.6–2.7: Nonhomogeneous ODEs and Solution Methods

Nonhomogeneous Linear ODEs

Nonhomogeneous ODEs include a nonzero right-hand side: $a_n y^{(n)} + \cdots + a_0 y = f(x)$.

• Undetermined Coefficients: Guess a form for the particular solution based on $f(x)$, substitute, and solve for coefficients.

• Variation of Parameters: Use the homogeneous solution and integrate to find a particular solution.

• Always solve the homogeneous equation first.

• If the guess for undetermined coefficients overlaps with the homogeneous solution, multiply by $x$ as needed.

General Solution Structure

• General solution: $y = y_h + y_p$, where $y_h$ is the complementary (homogeneous) solution and $y_p$ is the particular solution.

• For second-order linear ODEs: $y_h = C_1 y_1 + C_2 y_2$.

• Variation of Parameters formula: $y_p = y_1 \int \frac{y_2 f(x)}{W} dx - y_2 \int \frac{y_1 f(x)}{W} dx$ where $W$ is the Wronskian of $y_1$ and $y_2$.

Example

• Given $y'' + y = \sin x$, guess $y_p = A \sin x + B \cos x$ for undetermined coefficients.

Section 2.8: Model Equations and Notation

Modeling with ODEs

Model equations describe physical systems and their behavior over time. Understanding notation and parameters is essential for interpreting solutions.

• Model: $mx'' + cx' + kx = f(t)$, where $f(t)$ is an external force.

• Notation $x_h$ and $x_p$ refer to the complementary and particular solutions, respectively.

• Parameters $m$, $c$, $k$ must be identified and interpreted in context.

Example

• For a forced mass-spring system: $x'' + 2x' + x = \cos t$.