MIT 18.03 - Differential Equations & Laplace Transforms Flashcards

Question 1

Integrating Factor Method

Accepted Answer

Used for first-order linear ODEs $y' + P(x)y = Q(x)$. The integrating factor is $u(x) = e^{\int P(x) dx}$. Multiplying the entire ODE by $u(x)$ yields $\frac{d}{dx}[u(x)y] = u(x)Q(x)$.

Question 2

First-Order Separable ODE

Accepted Answer

An ODE of the form $\frac{dy}{dx} = g(x)h(y)$. Solved by isolating variables and integrating both sides: $\int \frac{1}{h(y)} dy = \int g(x) dx + C$.

Question 3

Exact ODEs & Potential Function

Accepted Answer

An equation $M(x,y)dx + N(x,y)dy = 0$ is exact if $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$. The solution is $f(x,y) = C$, where $\
abla f = \langle M, N \angle$.

Question 4

Second-Order Homogeneous ODE with Constant Coefficients

Accepted Answer

An equation $ay'' + by' + cy = 0$. Solved by substituting $y = e^{rt}$, which yields the characteristic equation $ar^2 + br + c = 0$.

Question 5

Characteristic Equation Roots & General Solutions

Accepted Answer

For $ar^2 + br + c = 0$:
- **Distinct real roots ($r_1 \
eq r_2$):** $y = c_1 e^{r_1 t} + c_2 e^{r_2 t}$
- **Repeated real root ($r_1 = r_2 = r$):** $y = (c_1 + c_2 t) e^{rt}$
- **Complex roots ($r = \alpha \pm i\beta$):** $y = e^{\alpha t}(c_1 \cos(\beta t) + c_2 \sin(\beta t))$

Question 6

Method of Undetermined Coefficients

Accepted Answer

Solves $ay'' + by' + cy = f(t)$ by proposing a trial solution $y_p(t)$ matching the form of $f(t)$ (polynomials, exponentials, sines/cosines). Multiply $y_p$ by $t^s$ if any term duplicates a homogeneous solution.

Question 7

Wronskian Matrix & Determinant

Accepted Answer

For functions $y_1, y_2$, $W(y_1, y_2)(t) = \det \begin{pmatrix} y_1 & y_2 \\ y_1' & y_2' \end{pmatrix} = y_1 y_2' - y_1' y_2$. If $W \
eq 0$, $\{y_1, y_2\}$ forms a fundamental set of linearly independent solutions.

Question 8

Variation of Parameters Formula

Accepted Answer

For $y'' + p(t)y' + q(t)y = f(t)$ with fundamental solutions $y_1, y_2$:
$y_p(t) = -y_1(t)\int \frac{y_2(t)f(t)}{W(y_1,y_2)} dt + y_2(t)\int \frac{y_1(t)f(t)}{W(y_1,y_2)} dt$

Question 9

Matrix Representation of Linear ODE Systems

Accepted Answer

A system of $n$ coupled first-order linear ODEs written as $\mathbf{x}'(t) = \mathbf{A}\mathbf{x}(t)$, where $\mathbf{A}$ is an $n \	imes n$ coefficient matrix and $\mathbf{x}(t)$ is a column vector of functions.

Question 10

General Solution to $\mathbf{x}' = \mathbf{A}\mathbf{x}$ (Real Distinct Eigenvalues)

Accepted Answer

If $\mathbf{A}$ has $n$ linearly independent eigenvectors $\mathbf{v}_i$ with distinct eigenvalues $\lambda_i$, the general solution is:
$\mathbf{x}(t) = \sum_{i=1}^n c_i \mathbf{v}_i e^{\lambda_i t}$

Question 11

Complex Eigenvalues in Linear Systems

Accepted Answer

For a real matrix with complex eigenvalue $\lambda = \alpha + i\beta$ and eigenvector $\mathbf{v} = \mathbf{a} + i\mathbf{b}$, the complex solution $\mathbf{x}_{comp}(t) = (\mathbf{a} + i\mathbf{b})e^{(\alpha + i\beta)t}$ expands via Euler's formula to produce two real basis solutions: $\	ext{Re}(\mathbf{x}_{comp})$ and $\	ext{Im}(\mathbf{x}_{comp})$.

Question 12

Definition of the Laplace Transform

Accepted Answer

The integral transform converting $f(t)$ from time domain to frequency domain $s$:
$\mathcal{L}\{f(t)\} = F(s) = \int_{0}^{\infty} e^{-st} f(t) dt$

Question 13

Laplace Transform of Derivatives

Accepted Answer

- $\mathcal{L}\{f'(t)\} = sF(s) - f(0)$
- $\mathcal{L}\{f''(t)\} = s^2 F(s) - s f(0) - f'(0)$
- Converts differential equations into algebraic equations.

Question 14

Common Laplace Transforms: Polynomials & Exponentials

Accepted Answer

- $\mathcal{L}\{1\} = \frac{1}{s}$
- $\mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}} \quad (n \in \mathbb{N})$
- $\mathcal{L}\{e^{at}\} = \frac{1}{s-a}$

Question 15

Common Laplace Transforms: Sine & Cosine

Accepted Answer

- $\mathcal{L}\{\cos(\omega t)\} = \frac{s}{s^2 + \omega^2}$
- $\mathcal{L}\{\sin(\omega t)\} = \frac{\omega}{s^2 + \omega^2}$

Question 16

First Shifting Theorem (s-shift)

Accepted Answer

$\mathcal{L}\{e^{at} f(t)\} = F(s - a)$
Multiplying $f(t)$ by an exponential shifts its Laplace transform in the $s$-domain by $a$ units.

Question 17

Heaviside Step Function & Second Shifting Theorem (t-shift)

Accepted Answer

The unit step function $u(t - a) = 1$ if $t \ge a$ and $0$ if $t < a$.
Second Shifting Theorem: $\mathcal{L}\{f(t - a) u(t - a)\} = e^{-as} F(s)$.

Question 18

Dirac Delta Function & Its Laplace Transform

Accepted Answer

Model of an ideal impulse at $t = a$. Key property: $\int_{0}^{\infty} f(t) \delta(t - a) dt = f(a)$.
Laplace transform: $\mathcal{L}\{\delta(t - a)\} = e^{-as}$.

Question 19

Convolution Theorem for Laplace Transforms

Accepted Answer

The convolution is defined as $(f * g)(t) = \int_{0}^{t} f(\	au) g(t - \	au) d\	au$.
Convolution Theorem: $\mathcal{L}\{(f * g)(t)\} = F(s) \cdot G(s)$.

Question 20

Resonance in Mechanical Oscillators

Accepted Answer

Occurs in $y'' + \omega_0^2 y = F_0 \cos(\omega t)$ when the driving frequency matches the natural frequency ($\omega = \omega_0$). The particular solution grows linearly over time: $y_p(t) = \frac{F_0}{2\omega_0} t \sin(\omega_0 t)$.

MIT 18.03 - Differential Equations & Laplace Transforms

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