MIT 18.03 - Differential Equations: Modeling & Solutions Flashcards

Question 1

Define the Laplace Transform of a function $f(t)$.

Accepted Answer

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

Question 2

What is an Ordinary Differential Equation (ODE)?

Accepted Answer

An equation involving an unknown function of one independent variable and its derivatives with respect to that variable.

Question 3

What is the order of an ODE?

Accepted Answer

The order of the highest derivative present in the equation.

Question 4

How do you solve a separable first-order ODE?

Accepted Answer

A separable ODE can be written in the form $g(y) dy = f(x) dx$. Integrate both sides to find the implicit or explicit solution.

Question 5

What is the standard form of a linear first-order ODE and how is it solved?

Accepted Answer

The standard form is $\frac{dy}{dx} + P(x)y = Q(x)$. It is solved using an integrating factor $I(x) = e^{\int P(x) dx}$. Multiply the ODE by $I(x)$, then integrate both sides.

Question 6

What is the condition for an ODE $M(x,y)dx + N(x,y)dy = 0$ to be exact?

Accepted Answer

The ODE is exact if $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$.

Question 7

How do you solve an exact ODE?

Accepted Answer

If $M(x,y)dx + N(x,y)dy = 0$ is exact, find a potential function $\phi(x,y)$ such that $\frac{\partial \phi}{\partial x} = M(x,y)$ and $\frac{\partial \phi}{\partial y} = N(x,y)$. The general solution is $\phi(x,y) = C$.

Question 8

What is the characteristic equation for a homogeneous second-order linear ODE with constant coefficients $ay'' + by' + cy = 0$?

Accepted Answer

The characteristic equation is $ar^2 + br + c = 0$.

Question 9

What is the general solution for $ay'' + by' + cy = 0$ when the characteristic equation has distinct real roots $r_1, r_2$?

Accepted Answer

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

Question 10

What is the general solution for $ay'' + by' + cy = 0$ when the characteristic equation has a repeated real root $r$?

Accepted Answer

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

Question 11

What is the general solution for $ay'' + by' + cy = 0$ when the characteristic equation has complex conjugate roots $\alpha \pm i\beta$?

Accepted Answer

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

Question 12

Explain the principle of superposition for linear homogeneous ODEs.

Accepted Answer

If $y_1$ and $y_2$ are solutions to a linear homogeneous ODE, then any linear combination $C_1y_1 + C_2y_2$ is also a solution.

Question 13

What is the general solution structure for a non-homogeneous linear ODE $L[y] = g(x)$?

Accepted Answer

The general solution is $y(x) = y_h(x) + y_p(x)$, where $y_h$ is the general solution to the associated homogeneous equation $L[y]=0$, and $y_p$ is any particular solution to $L[y]=g(x)$.

Question 14

Briefly describe the method of Undetermined Coefficients.

Accepted Answer

This method finds a particular solution $y_p$ for non-homogeneous linear ODEs with constant coefficients when $g(x)$ is a polynomial, exponential, sine/cosine, or a combination. One guesses the form of $y_p$ based on $g(x)$ and its derivatives, then substitutes it into the ODE to find the coefficients.

Question 15

When using Undetermined Coefficients, what adjustment is needed if the initial guess for $y_p$ is a solution to the homogeneous equation?

Accepted Answer

Multiply the initial guess by $x$ (or $x^2$ if $x$ times the guess is also a homogeneous solution) until it is no longer a solution to the associated homogeneous equation.

Question 16

Briefly describe the method of Variation of Parameters for $y'' + p(x)y' + q(x)y = f(x)$.

Accepted Answer

A general method for finding a particular solution $y_p = u_1(x)y_1(x) + u_2(x)y_2(x)$, where $y_1, y_2$ are linearly independent solutions to the homogeneous equation. The functions $u_1', u_2'$ are found using formulas involving the Wronskian $W(y_1, y_2) = y_1y_2' - y_2y_1'$ and $f(x)$.

Question 17

How does the Laplace Transform simplify solving ODEs?

Accepted Answer

It transforms a linear ODE with constant coefficients from a differential equation in the $t$-domain into an algebraic equation in the $s$-domain. Solving for $F(s)$ and then applying the inverse Laplace Transform yields the solution $f(t)$.

Question 18

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

Accepted Answer

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

Question 19

What is a phase plane in the context of systems of ODEs?

Accepted Answer

A 2D plane where the axes represent the dependent variables (e.g., $x$ and $y$ for a system $\frac{dx}{dt}=F(x,y), \frac{dy}{dt}=G(x,y)$). Solutions are represented as trajectories or phase paths in this plane, showing the system's behavior over time.

Question 20

What are critical points (or equilibrium points) in a phase plane?

Accepted Answer

Points $(x_0, y_0)$ where $\frac{dx}{dt} = 0$ and $\frac{dy}{dt} = 0$ simultaneously. At these points, the system is at rest, and trajectories do not move.

Question 21

Name and briefly describe two types of critical points based on stability.

Accepted Answer

1. **Stable Node:** All nearby trajectories approach the critical point as $t 	o \infty$. 
2. **Unstable Saddle Point:** Trajectories approach the critical point along specific directions but move away along others. 
3. **Stable Spiral:** Trajectories spiral inwards towards the critical point. 
4. **Center:** Trajectories form closed loops around the critical point (stable, but not asymptotically stable).

Question 22

What are nullclines in a phase plane analysis?

Accepted Answer

Curves where one of the derivatives is zero. For a system $\frac{dx}{dt}=F(x,y)$ and $\frac{dy}{dt}=G(x,y)$, the $x$-nullcline is where $F(x,y)=0$, and the $y$-nullcline is where $G(x,y)=0$. Critical points are the intersections of nullclines.

Question 23

How are eigenvalues and eigenvectors used to analyze linear systems of ODEs $\mathbf{x}' = A\mathbf{x}$?

Accepted Answer

Eigenvalues of the matrix $A$ determine the stability and type of the critical point at the origin (e.g., node, saddle, spiral, center). Eigenvectors define the directions of trajectories, especially for real eigenvalues, showing how solutions move towards or away from the critical point.

MIT 18.03 - Differential Equations: Modeling & Solutions

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