MIT 18.03 - Differential Equations, Fourier Series & Laplace Transforms Flashcards

Question 1

If the characteristic equation $ar^2 + br + c = 0$ has two distinct real roots $r_1$ and $r_2$, what is the general solution to $ay'' + by' + cy = 0$?

Accepted Answer

The general solution is $y(x) = C_1e^{r_1x} + C_2e^{r_2x}$.

Question 2

What is the general form of a first-order linear ordinary differential equation (ODE)?

Accepted Answer

A first-order linear ODE has the form $y' + p(x)y = q(x)$, where $p(x)$ and $q(x)$ are continuous functions of $x$.

Question 3

How do you solve a first-order linear ODE $y' + p(x)y = q(x)$ using an integrating factor?

Accepted Answer

Multiply the entire equation by the integrating factor $\mu(x) = e^{\int p(x) dx}$. The left side becomes $(\mu(x)y)'$, so $rac{d}{dx}(\mu(x)y) = \mu(x)q(x)$. Integrate both sides: $\mu(x)y = \int \mu(x)q(x) dx + C$, then solve for $y$.

Question 4

For a second-order homogeneous linear ODE $ay'' + by' + cy = 0$, what is the characteristic equation?

Accepted Answer

The characteristic equation is $ar^2 + br + c = 0$. Its roots determine the form of the general solution.

Question 5

If the characteristic equation $ar^2 + br + c = 0$ has a repeated real root $r$, what is the general solution to $ay'' + by' + cy = 0$?

Accepted Answer

The general solution is $y(x) = C_1e^{rx} + C_2xe^{rx}$.

Question 6

If the characteristic equation $ar^2 + br + c = 0$ has complex conjugate roots $r = \alpha \pm ieta$, what is the general solution to $ay'' + by' + cy = 0$?

Accepted Answer

The general solution is $y(x) = e^{\alpha x}(C_1\cos(eta x) + C_2\sin(eta x))$.

Question 7

Briefly explain the Method of Undetermined Coefficients for solving $ay'' + by' + cy = f(x)$.

Accepted Answer

This method finds a particular solution $y_p(x)$ by guessing a form similar to $f(x)$ (e.g., if $f(x)$ is a polynomial, guess a polynomial of the same degree; if $f(x)$ is $e^{kx}$, guess $Ae^{kx}$). The coefficients in the guess are then determined by substituting $y_p$ into the ODE. If the guess duplicates a term in the homogeneous solution, multiply by $x$ (or $x^2$).

Question 8

State the formula for a particular solution $y_p(x)$ using Variation of Parameters for $y'' + p(x)y' + q(x)y = f(x)$, given homogeneous solutions $y_1, y_2$.

Accepted Answer

Let $W(y_1, y_2) = y_1y_2' - y_2y_1'$ be the Wronskian. Then $y_p(x) = -y_1(x)\int rac{y_2(x)f(x)}{W(y_1, y_2)} dx + y_2(x)\int rac{y_1(x)f(x)}{W(y_1, y_2)} dx$.

Question 9

How is a system of first-order linear ODEs typically written in matrix form?

Accepted Answer

A system of $n$ first-order linear ODEs can be written as $\mathbf{x}'(t) = A\mathbf{x}(t)$, where $\mathbf{x}(t)$ is a vector of dependent variables, $\mathbf{x}'(t)$ is its derivative, and $A$ is an $n 	imes n$ constant matrix.

Question 10

If matrix $A$ has distinct real eigenvalues $\lambda_1, \dots, \lambda_n$ with corresponding eigenvectors $\mathbf{v}_1, \dots, \mathbf{v}_n$, what is the general solution to $\mathbf{x}'(t) = A\mathbf{x}(t)$?

Accepted Answer

The general solution is $\mathbf{x}(t) = C_1\mathbf{v}_1e^{\lambda_1 t} + C_2\mathbf{v}_2e^{\lambda_2 t} + \dots + C_n\mathbf{v}_ne^{\lambda_n t}$.

Question 11

Define the matrix exponential $e^{At}$ and state its use in solving $\mathbf{x}'(t) = A\mathbf{x}(t)$ with initial condition $\mathbf{x}(0) = \mathbf{x}_0$.

Accepted Answer

The matrix exponential is defined by the series $e^{At} = I + At + rac{(At)^2}{2!} + rac{(At)^3}{3!} + \dots = \sum_{k=0}^{\infty} rac{(At)^k}{k!}$. The solution to the initial value problem is $\mathbf{x}(t) = e^{At}\mathbf{x}_0$.

Question 12

Define the Laplace Transform $\mathcal{L}\{f(t)\}$ of a function $f(t)$ for $t \ge 0$.

Accepted Answer

The Laplace Transform is defined as $\mathcal{L}\{f(t)\} = F(s) = \int_0^{\infty} e^{-st}f(t) dt$, for values of $s$ where the integral converges.

Question 13

State the Laplace Transform of the first derivative $f'(t)$.

Accepted Answer

$\mathcal{L}\{f'(t)\} = sF(s) - f(0)$, where $F(s) = \mathcal{L}\{f(t)\}$.

Question 14

State the Laplace Transform of the second derivative $f''(t)$.

Accepted Answer

$\mathcal{L}\{f''(t)\} = s^2F(s) - sf(0) - f'(0)$, where $F(s) = \mathcal{L}\{f(t)\}$.

Question 15

What is the Laplace Transform of $e^{at}$?

Accepted Answer

$\mathcal{L}\{e^{at}\} = rac{1}{s-a}$, for $s > a$.

Question 16

What is the Laplace Transform of $\sin(at)$?

Accepted Answer

$\mathcal{L}\{\sin(at)\} = rac{a}{s^2+a^2}$, for $s > 0$.

Question 17

What is the Laplace Transform of the Heaviside step function $u(t-a)$?

Accepted Answer

$\mathcal{L}\{u(t-a)\} = rac{e^{-as}}{s}$, for $s > 0$.

Question 18

What is the Laplace Transform of the Dirac delta function $\delta(t-a)$?

Accepted Answer

$\mathcal{L}\{\delta(t-a)\} = e^{-as}$.

Question 19

State the Convolution Theorem for Laplace Transforms.

Accepted Answer

If $\mathcal{L}\{f(t)\} = F(s)$ and $\mathcal{L}\{g(t)\} = G(s)$, then $\mathcal{L}\{(f*g)(t)\} = F(s)G(s)$, where $(f*g)(t) = \int_0^t f(	au)g(t-	au) d	au$.

Question 20

Define the Fourier Series for a $2L$-periodic function $f(x)$ on the interval $[-L, L]$.

Accepted Answer

The Fourier Series is given by $f(x) = a_0 + \sum_{n=1}^{\infty} (a_n\cos(rac{n\pi x}{L}) + b_n\sin(rac{n\pi x}{L}))$.

Question 21

Provide the formulas for the Fourier coefficients $a_0, a_n, b_n$ for a function $f(x)$ on $[-L, L]$.

Accepted Answer

The coefficients are:
*   $a_0 = rac{1}{2L}\int_{-L}^{L} f(x) dx$
*   $a_n = rac{1}{L}\int_{-L}^{L} f(x)\cos(rac{n\pi x}{L}) dx$
*   $b_n = rac{1}{L}\int_{-L}^{L} f(x)\sin(rac{n\pi x}{L}) dx$

Question 22

When is a Fourier Sine Series used, and what is its form for a function $f(x)$ on $[0, L]$?

Accepted Answer

A Fourier Sine Series is used for odd functions or to represent a function on $[0, L]$ by extending it oddly to $[-L, L]$. It consists only of sine terms: $f(x) = \sum_{n=1}^{\infty} b_n\sin(rac{n\pi x}{L})$, where $b_n = rac{2}{L}\int_{0}^{L} f(x)\sin(rac{n\pi x}{L}) dx$.

Question 23

When is a Fourier Cosine Series used, and what is its form for a function $f(x)$ on $[0, L]$?

Accepted Answer

A Fourier Cosine Series is used for even functions or to represent a function on $[0, L]$ by extending it evenly to $[-L, L]$. It consists only of cosine terms: $f(x) = a_0 + \sum_{n=1}^{\infty} a_n\cos(rac{n\pi x}{L})$, where $a_0 = rac{1}{L}\int_{0}^{L} f(x) dx$ and $a_n = rac{2}{L}\int_{0}^{L} f(x)\cos(rac{n\pi x}{L}) dx$.

MIT 18.03 - Differential Equations, Fourier Series & Laplace Transforms

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