Differential Equations

Modeling forced mechanical vibrations using second-order non-homogeneous linear differential equations.

Derive the particular solution $x_p(t)$ for a forced damped oscillation given the differential equation $m x'' + c x' + k x = F_0 \cos(\omega t)$ and express it in terms of a single cosine term using the phase angle.

Using the heat equation, determine how the temperature distribution along a one-dimensional rod changes over time given certain boundary conditions and an initial temperature function.

Using the heat equation, determine how the temperature distribution along a one-dimensional rod changes over time given certain boundary conditions and an initial temperature function.

Use the Fourier Transform to solve the 1D heat equation $U_T = \alpha^2 U_{XX}$, assuming an infinite 1D piece of metal with an initial temperature distribution $U(X, 0)$ that dies out to zero at $\pm\infty$.

Given a material bar with ends kept at temperature zero, analyze the heat flow and temperature distribution over time using the heat equation.

Suppose we have a metal bar of length $L$. The ends of the bar are attached to some heat baths, so their temperatures are fixed. The center of the bar is heated and becomes very hot. Describe how the temperature distribution changes and eventually stabilizes over time as the bar cools off using Fourier series.

Find the fundamental matrix for the system $X' = \begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix} X$.

Find a fundamental matrix for a given constant coefficient ODE system, which you have solved before in the last couple sections.

Consider the system $X' = AX$, where $A = \begin{pmatrix} 5 & -1 \\ -2 & 1 \end{pmatrix}$. Find a fundamental matrix $\Phi(t)$ such that $\Phi(0)$ is the identity matrix and use it to solve the initial value problem with $X(0) = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$.

Solve the Bernoulli differential equation: $y' - 5y = -\frac{5}{2}x y^3$ using the substitution $u = y^{-2}$ and the method of integrating factors.

Using the Bernoulli equation, solve the differential equation of the form: $y' + P(t)y = G(t)y^n$ by introducing a new variable $V$ and applying the integrating factor method.

For the Bernoulli differential equation $\frac{dy}{dx} + p(x)y = q(x) y^n$, manipulate it to get rid of the $y^n$ term by multiplying both sides by $y^{-n}$, and solve for $y$ using the substitution $u = y^{1-n}$.

Solve the Bernoulli differential equation $\frac{dy}{dx} + p(x) y = q(x) y^2$ by multiplying both sides by $y^{-2}$ and performing a variable substitution.

Solve the differential equation by dividing both sides by $y$ and integrating, then using the transformation $u = \frac{1}{y}$ to solve the equation.

Solve the Bernoulli differential equation: rac{dy}{dx} - y = e^{2x} y^3

dy/dt = ky

Solve a differential equation of the form $\frac{dy}{dt} = ky$ using separation of variables and integrating both sides to find $y = y_0 e^{kt}$.

Using the factorization technique, solve the differential equation rac{dy}{dx} = y + ky.