MIT 18.03 - Ordinary Differential Equations & Laplace Transforms Flashcards

Question 1

What is the general form of a second-order linear homogeneous ODE with constant coefficients?

Accepted Answer

The general form is $a y'' + b y' + c y = 0$, where $a, b, c$ are constants and $a 
eq 0$.

Question 2

What is an exact differential equation?

Accepted Answer

A first-order ODE $M(x,y)dx + N(x,y)dy = 0$ is exact if $rac{\partial M}{\partial y} = rac{\partial N}{\partial x}$. The solution is given by $F(x,y) = C$, where $rac{\partial F}{\partial x} = M$ and $rac{\partial F}{\partial y} = N$.

Question 3

What is an Ordinary Differential Equation (ODE)?

Accepted Answer

An Ordinary Differential Equation (ODE) is an equation involving an unknown function of a single independent variable and its derivatives with respect to that variable. For example, $rac{dy}{dx} = xy$.

Question 4

How is the order of an ODE determined?

Accepted Answer

The order of an ODE is determined by the highest derivative present in the equation. For example, $y''' + 2y'' - y' + 5y = \sin(x)$ is a third-order ODE.

Question 5

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

Accepted Answer

The general form of a first-order linear ODE is $rac{dy}{dx} + P(x)y = Q(x)$, where $P(x)$ and $Q(x)$ are continuous functions of $x$.

Question 6

How do you solve a first-order linear ODE using an integrating factor?

Accepted Answer

To solve $rac{dy}{dx} + P(x)y = Q(x)$, first calculate the integrating factor $I(x) = e^{\int P(x)dx}$. Then, multiply the entire ODE by $I(x)$ to get $rac{d}{dx}[I(x)y] = I(x)Q(x)$. Integrate both sides with respect to $x$ to find $y$: $I(x)y = \int I(x)Q(x)dx + C$, so $y = rac{1}{I(x)}\left(\int I(x)Q(x)dx + Cight)$.

Question 7

What is a separable ODE and how is it solved?

Accepted Answer

A separable ODE is a first-order ODE that can be written in the form $rac{dy}{dx} = g(x)h(y)$. It is solved by separating the variables and integrating: $\int rac{1}{h(y)}dy = \int g(x)dx$.

Question 8

How is the characteristic equation derived for $a y'' + b y' + c y = 0$?

Accepted Answer

Assuming a solution of the form $y = e^{rx}$, we substitute $y' = re^{rx}$ and $y'' = r^2e^{rx}$ into the ODE. This yields $a r^2 e^{rx} + b r e^{rx} + c e^{rx} = 0$. Dividing by $e^{rx}$ (which is never zero), we get the characteristic equation: $a r^2 + b r + c = 0$.

Question 9

What is the general solution for $a y'' + b y' + c y = 0$ when the characteristic equation has 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 10

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

Accepted Answer

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

Question 11

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

Accepted Answer

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

Question 12

When is the method of undetermined coefficients used, and what is its general approach?

Accepted Answer

It's used for non-homogeneous linear ODEs $a y'' + b y' + c y = g(x)$ when $g(x)$ is a combination of polynomials, exponentials, sines, or cosines. The approach is to guess a particular solution $y_p$ based on the form of $g(x)$ and its derivatives, then substitute it into the ODE to find the coefficients. The general solution is $y = y_h + y_p$, where $y_h$ is the homogeneous solution.

Question 13

When is the method of variation of parameters preferred over undetermined coefficients for non-homogeneous ODEs?

Accepted Answer

Variation of parameters is a more general method that works for any continuous $g(x)$ and even for ODEs with non-constant coefficients, provided the homogeneous solution is known. It is preferred when $g(x)$ is not of the 'undetermined coefficients' form (e.g., $	an(x)$ or $\ln(x)$).

Question 14

What is the general form of a homogeneous system of linear first-order ODEs?

Accepted Answer

The general form is $\mathbf{x}' = A\mathbf{x}$, where $\mathbf{x}$ is a vector of unknown functions, $\mathbf{x}'$ is its derivative, and $A$ is a constant matrix.

Question 15

How are eigenvalues and eigenvectors used to solve systems of linear ODEs?

Accepted Answer

For $\mathbf{x}' = A\mathbf{x}$, if $A$ has distinct eigenvalues $\lambda_i$ with corresponding eigenvectors $\mathbf{v}_i$, the general solution is $\mathbf{x}(t) = C_1 \mathbf{v}_1 e^{\lambda_1 t} + C_2 \mathbf{v}_2 e^{\lambda_2 t} + \dots + C_n \mathbf{v}_n e^{\lambda_n t}$.

Question 16

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

Accepted Answer

The Laplace Transform of a function $f(t)$ for $t \ge 0$ is defined as $\mathcal{L}\{f(t)\} = F(s) = \int_{0}^{\infty} e^{-st} f(t) dt$, provided the integral converges. Here, $s$ is a complex variable.

Question 17

What is the Inverse Laplace Transform?

Accepted Answer

The Inverse Laplace Transform, denoted $\mathcal{L}^{-1}\{F(s)\} = f(t)$, is the operation that recovers the original function $f(t)$ from its Laplace Transform $F(s)$. It's often found using tables or partial fraction decomposition.

Question 18

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

Accepted Answer

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

Question 19

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

Accepted Answer

The Laplace Transform of the second derivative is $\mathcal{L}\{f''(t)\} = s^2F(s) - sf(0) - f'(0)$, where $F(s) = \mathcal{L}\{f(t)\}$ and $f(0), f'(0)$ are initial values.

Question 20

What is $\mathcal{L}\{e^{at}\}$?

Accepted Answer

The Laplace Transform of $e^{at}$ is $\mathcal{L}\{e^{at}\} = rac{1}{s-a}$, for $s > a$.

Question 21

What is $\mathcal{L}\{\sin(kt)\}$?

Accepted Answer

The Laplace Transform of $\sin(kt)$ is $\mathcal{L}\{\sin(kt)\} = rac{k}{s^2 + k^2}$, for $s > 0$.

Question 22

Briefly explain the steps to solve an initial value problem (IVP) using Laplace Transforms.

Accepted Answer

1. Take the Laplace Transform of both sides of the ODE, using initial conditions. 2. Solve the resulting algebraic equation for $F(s) = \mathcal{L}\{y(t)\}$. 3. Find the inverse Laplace Transform $\mathcal{L}^{-1}\{F(s)\}$ to obtain the solution $y(t)$.

Question 23

What is the Heaviside step function (Unit Step Function) and its Laplace Transform?

Accepted Answer

The Heaviside step function, $u_c(t)$, is defined as $u_c(t) = 0$ for $t < c$ and $u_c(t) = 1$ for $t \ge c$. Its Laplace Transform is $\mathcal{L}\{u_c(t)\} = rac{e^{-cs}}{s}$.

Question 24

What is the Dirac Delta function and its Laplace Transform?

Accepted Answer

The Dirac Delta function, $\delta(t-c)$, is a generalized function (not a true function) that is zero everywhere except at $t=c$, and its integral over all real numbers is 1. It models an impulse. Its Laplace Transform is $\mathcal{L}\{\delta(t-c)\} = e^{-cs}$ for $c \ge 0$.

MIT 18.03 - Ordinary Differential Equations & Laplace Transforms

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