MIT 18.03 - Differential Equations: Linear Systems, Phase Planes & Laplace Transforms Flashcards

Question 1

What is the general solution for $a y'' + b y' + c y = 0$ if the characteristic equation has a repeated real root $r$?

Accepted Answer

The general solution is $y(x) = C_1 e^{rx} + C_2 x e^{rx}$.

Question 2

What is the general solution for $a y'' + b y' + c y = 0$ if the characteristic equation has two distinct real roots $r_1, r_2$?

Accepted Answer

The general solution is $y(x) = C_1 e^{r_1 x} + C_2 e^{r_2 x}$.

Question 3

What is the general form of a first-order linear differential equation?

Accepted Answer

A first-order linear differential equation can be written in the form $\frac{dy}{dx} + P(x)y = Q(x)$.

Question 4

What is the integrating factor formula $\mu(x)$ used to solve first-order linear ODEs?

Accepted Answer

The integrating factor is given by $\mu(x) = e^{\int P(x)dx}$. Multiplying the ODE by $\mu(x)$ allows the left side to be written as $\frac{d}{dx}(\mu(x)y)$.

Question 5

What is the characteristic equation for a second-order homogeneous ODE $a y'' + b y' + c y = 0$?

Accepted Answer

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

Question 6

What is the general solution for $a y'' + b y' + c y = 0$ if the characteristic equation has complex conjugate roots $r = \alpha \pm i\beta$?

Accepted Answer

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

Question 7

What is the definition of the Laplace Transform of a function $f(t)$?

Accepted Answer

The Laplace Transform $\mathcal{L}\{f(t)\}$ 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 8

What is the Laplace Transform of the derivative $f'(t)$?

Accepted Answer

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

Question 9

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

Accepted Answer

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

Question 10

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

Accepted Answer

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

Question 11

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

Accepted Answer

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

Question 12

How do you represent a system of first-order linear ODEs in matrix form?

Accepted Answer

A system like $\frac{dx_1}{dt} = a_{11}x_1 + a_{12}x_2$, $\frac{dx_2}{dt} = a_{21}x_1 + a_{22}x_2$ can be written as $\mathbf{x}' = A\mathbf{x}$, where $\mathbf{x} = \begin{pmatrix} x_1 \ x_2 \end{pmatrix}$ and $A = \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{pmatrix}$.

Question 13

How do you find the eigenvalues of a matrix $A$?

Accepted Answer

Eigenvalues $\lambda$ are found by solving the characteristic equation $\det(A - \lambda I) = 0$, where $I$ is the identity matrix.

Question 14

How do you find the eigenvectors corresponding to an eigenvalue $\lambda$?

Accepted Answer

Eigenvectors $\mathbf{v}$ are non-zero vectors that satisfy the equation $(A - \lambda I)\mathbf{v} = \mathbf{0}$.

Question 15

What type of critical point in a phase plane corresponds to real, distinct eigenvalues of opposite signs?

Accepted Answer

A Saddle Point. It is always unstable.

Question 16

What type of critical point in a phase plane corresponds to real, distinct eigenvalues of the same sign?

Accepted Answer

A Node. It is stable (asymptotically stable) if both eigenvalues are negative, and unstable if both are positive.

Question 17

What type of critical point in a phase plane corresponds to complex conjugate eigenvalues with a non-zero real part ($\alpha \pm i\beta$, $\alpha 
eq 0$)?

Accepted Answer

A Spiral (or Focus). It is stable (asymptotically stable) if $\alpha < 0$ and unstable if $\alpha > 0$.

Question 18

What type of critical point in a phase plane corresponds to purely imaginary eigenvalues ($\pm i\beta$)?

Accepted Answer

A Center. It is stable but not asymptotically stable. Trajectories are closed ellipses around the critical point.

Question 19

What is the general solution for a system $\mathbf{x}' = A\mathbf{x}$ if $A$ has two distinct real eigenvalues $\lambda_1, \lambda_2$ with corresponding eigenvectors $\mathbf{v}_1, \mathbf{v}_2$?

Accepted Answer

The general solution is $\mathbf{x}(t) = C_1 e^{\lambda_1 t} \mathbf{v}_1 + C_2 e^{\lambda_2 t} \mathbf{v}_2$.

Question 20

What is the general solution for a system $\mathbf{x}' = A\mathbf{x}$ if $A$ has complex conjugate eigenvalues $\lambda = \alpha \pm i\beta$ with corresponding eigenvector $\mathbf{v}$ for $\alpha + i\beta$?

Accepted Answer

If $\mathbf{v} = \mathbf{a} + i\mathbf{b}$, then the general solution is $\mathbf{x}(t) = C_1 e^{\alpha t}(\mathbf{a} \cos(\beta t) - \mathbf{b} \sin(\beta t)) + C_2 e^{\alpha t}(\mathbf{a} \sin(\beta t) + \mathbf{b} \cos(\beta t))$.

Question 21

What is the principle of superposition for linear homogeneous ODEs?

Accepted Answer

If $y_1(x)$ and $y_2(x)$ are solutions to a linear homogeneous ODE, then any linear combination $C_1 y_1(x) + C_2 y_2(x)$ is also a solution. This principle extends to systems of linear ODEs.

Question 22

How does the Method of Undetermined Coefficients work for non-homogeneous ODEs?

Accepted Answer

For $a y'' + b y' + c y = g(x)$, first find the general solution to the homogeneous equation $y_h$. Then, guess a particular solution $y_p$ based on the form of $g(x)$ (e.g., polynomial, exponential, sine/cosine) and substitute it into the original ODE to find its coefficients. The general solution is $y = y_h + y_p$.

Question 23

What is the definition of an asymptotically stable critical point?

Accepted Answer

A critical point is asymptotically stable if all trajectories starting sufficiently close to it approach the critical point as $t 	o \infty$. This implies both stability and attraction to the equilibrium.

Question 24

When is an equilibrium point for a linear system $\mathbf{x}' = A\mathbf{x}$ considered stable?

Accepted Answer

An equilibrium point is stable if, for any small perturbation, the trajectories remain close to the equilibrium point for all future time. It is asymptotically stable if trajectories also approach the equilibrium point. This occurs when all eigenvalues of $A$ have non-positive real parts, and any eigenvalues with zero real parts are simple (not repeated).

Question 25

What is the Laplace Transform of $t^n$ (for non-negative integer $n$)?

Accepted Answer

$\mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}}$, for $s > 0$.

MIT 18.03 - Differential Equations: Linear Systems, Phase Planes & Laplace Transforms

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