MIT 18.03 - Differential Equations: Ordinary Differential Equations, Laplace & Fourier Flashcards

Question 1

Integrating Factor Method for First-Order Linear ODEs

Accepted Answer

For an ODE in standard form $y' + p(x)y = q(x)$, multiply by the integrating factor $u(x) = e^{\int p(x)dx}$. This transforms the LHS into $\frac{d}{dx}[u(x)y] = u(x)q(x)$, which integrates to $y(x) = \frac{1}{u(x)} \left( \int u(x)q(x)dx + C \ight)$.

Question 2

Separation of Variables Method

Accepted Answer

Applies to first-order ODEs of the form $\frac{dy}{dx} = g(x)h(y)$. Rewrite the equation as $\frac{1}{h(y)} dy = g(x) dx$ and integrate both sides directly: $\int \frac{1}{h(y)} dy = \int g(x) dx + C$.

Question 3

Characteristic Equation for Constant Coefficient Linear ODEs

Accepted Answer

For $a y'' + b y' + c y = 0$, substitute $y = e^{rt}$ to get $a r^2 + b r + c = 0$.
- **Distinct real roots ($r_1 \
eq r_2$):** $y(t) = c_1 e^{r_1 t} + c_2 e^{r_2 t}$
- **Repeated root ($r_1 = r_2 = r$):** $y(t) = c_1 e^{rt} + c_2 t e^{rt}$
- **Complex roots ($r = \alpha \pm i\beta$):** $y(t) = e^{\alpha t}(c_1 \cos \beta t + c_2 \sin \beta t)$

Question 4

Method of Undetermined Coefficients

Accepted Answer

A technique to find a particular solution $y_p(t)$ for $a y'' + b y' + c y = f(t)$ when $f(t)$ is a polynomial, exponential, sine, or cosine. Guess a trial solution matching $f(t)$, multiplying by $t^s$ (where $s$ is the multiplicity) if any term duplicates a solution to the homogeneous equation.

Question 5

Variation of Parameters Formula (2nd Order)

Accepted Answer

For $y'' + p(t)y' + q(t)y = f(t)$ with fundamental homogeneous solutions $y_1, y_2$, the particular solution is:
$y_p(t) = -y_1(t) \int \frac{y_2(t)f(t)}{W(y_1, y_2)(t)} dt + y_2(t) \int \frac{y_1(t)f(t)}{W(y_1, y_2)(t)} dt$
where $W(y_1, y_2)$ is the Wronskian.

Question 6

Wronskian Determinant $W(y_1, y_2)$

Accepted Answer

Defined as $W(y_1, y_2)(t) = y_1(t)y_2'(t) - y_1'(t)y_2(t)$. If $W(y_1, y_2)(t) \
eq 0$ at a point in an interval, $y_1$ and $y_2$ are linearly independent and form a fundamental set of solutions.

Question 7

Pure Resonance in Undamped Systems

Accepted Answer

Occurs in $m y'' + k y = F_0 \cos(\omega t)$ when driving frequency matches natural frequency ($\omega = \omega_0 = \sqrt{k/m}$). The particular solution grows linearly over time: $y_p(t) = \frac{F_0}{2 m \omega_0} t \sin(\omega_0 t)$, producing unbounded oscillations.

Question 8

Practical Resonance in Damped Systems

Accepted Answer

In damped forced systems $m y'' + b y' + k y = F_0 \cos(\omega t)$, steady-state amplitude $A(\omega) = \frac{F_0}{\sqrt{(k - m\omega^2)^2 + (b\omega)^2}}$ reaches a finite maximum at the resonant frequency $\omega_r = \sqrt{\omega_0^2 - \frac{b^2}{2m^2}}$ (provided $b < m\omega_0\sqrt{2}$).

Question 9

Definition of the Laplace Transform

Accepted Answer

$\mathcal{L}\{f(t)\} = F(s) = \int_{0}^{\infty} e^{-st} f(t) \, dt$. It maps time-domain differential equations $t \ge 0$ into algebraic equations in the complex frequency domain $s$.

Question 10

Laplace Transform of Derivatives

Accepted Answer

Converts differentiation in time to algebraic multiplication in $s$:
- $\mathcal{L}\{f'(t)\} = s F(s) - f(0)$
- $\mathcal{L}\{f''(t)\} = s^2 F(s) - s f(0) - f'(0)$
- $\mathcal{L}\{f^{(n)}(t)\} = s^n F(s) - \sum_{k=1}^n s^{n-k} f^{(k-1)}(0)$

Question 11

Fundamental Laplace Transform Pairs

Accepted Answer

- $\mathcal{L}\{1\} = \frac{1}{s}$
- $\mathcal{L}\{e^{at}\} = \frac{1}{s-a}$
- $\mathcal{L}\{\sin(\omega t)\} = \frac{\omega}{s^2 + \omega^2}$
- $\mathcal{L}\{\cos(\omega t)\} = \frac{s}{s^2 + \omega^2}$
- $\mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}}$

Question 12

Heaviside Step and Shift Theorems

Accepted Answer

- **First Shifting Theorem ($s$-shift):** $\mathcal{L}\{e^{at}f(t)\} = F(s-a)$
- **Second Shifting Theorem ($t$-shift):** $\mathcal{L}\{u(t-a)f(t-a)\} = e^{-as}F(s)$, where $u(t-a)$ is the Heaviside step function centered at $t = a$.

Question 13

Dirac Delta Function $\delta(t-a)$

Accepted Answer

Generalized function satisfying $\delta(t-a) = 0$ for $t \
eq a$ and $\int_{-\infty}^{\infty} \delta(t-a) dt = 1$. Its Laplace transform is $\mathcal{L}\{\delta(t-a)\} = e^{-as}$. It acts as the derivative of the Heaviside unit step function.

Question 14

Convolution Theorem for Laplace Transforms

Accepted Answer

$\mathcal{L}\{(f * g)(t)\} = F(s)G(s)$, where $(f * g)(t) = \int_{0}^{t} f(\	au)g(t - \	au) \, d\	au$. Multiplication in the $s$-domain corresponds to convolution in the time domain.

Question 15

Transfer Function $H(s)$ and Impulse Response $h(t)$

Accepted Answer

For an LTI system with input $X(s)$ and output $Y(s)$, the **Transfer Function** is $H(s) = \frac{Y(s)}{X(s)}$. The **Impulse Response** is $h(t) = \mathcal{L}^{-1}\{H(s)\}$, representing system output when input is $\delta(t)$.

Question 16

Real Fourier Series Representation ($2L$-Periodic)

Accepted Answer

Represent $f(t)$ on $[-L, L]$ as: $f(t) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos\left(\frac{n\pi t}{L}\ight) + b_n \sin\left(\frac{n\pi t}{L}\ight) \ight)$

Question 17

Euler-Fourier Coefficient Formulas

Accepted Answer

For a $2L$-periodic function:
- $a_0 = \frac{1}{L} \int_{-L}^{L} f(t) \, dt$
- $a_n = \frac{1}{L} \int_{-L}^{L} f(t) \cos\left(\frac{n\pi t}{L}\ight) dt$
- $b_n = \frac{1}{L} \int_{-L}^{L} f(t) \sin\left(\frac{n\pi t}{L}\ight) dt$

Question 18

Complex Exponential Fourier Series

Accepted Answer

$f(t) = \sum_{n=-\infty}^{\infty} c_n e^{i \frac{n\pi t}{L}}$, where coefficients are $c_n = \frac{1}{2L} \int_{-L}^{L} f(t) e^{-i \frac{n\pi t}{L}} dt$. The real coefficients map as $c_n = \frac{a_n - i b_n}{2}$ for $n > 0$.

Question 19

State Space System Representation

Accepted Answer

Converts an $n$-th order linear ODE into a system of first-order vector differential equations: $\mathbf{x}'(t) = \mathbf{A}\mathbf{x}(t) + \mathbf{f}(t)$, where $\mathbf{x}(t)$ is an $n \	imes 1$ state vector and $\mathbf{A}$ is an $n \	imes n$ system matrix.

Question 20

Matrix Exponential $e^{\mathbf{A}t}$

Accepted Answer

Defined by the power series $e^{\mathbf{A}t} = \sum_{k=0}^{\infty} \frac{\mathbf{A}^k t^k}{k!} = \mathbf{I} + \mathbf{A}t + \frac{\mathbf{A}^2 t^2}{2!} + \dots$. The unique solution to $\mathbf{x}' = \mathbf{A}\mathbf{x}$ with $\mathbf{x}(0) = \mathbf{x}_0$ is $\mathbf{x}(t) = e^{\mathbf{A}t}\mathbf{x}_0$.

Question 21

Solving $\mathbf{x}' = \mathbf{A}\mathbf{x}$ via Eigenvalues/Eigenvectors

Accepted Answer

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

MIT 18.03 - Differential Equations: Ordinary Differential Equations, Laplace & Fourier

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