Differential Equations

Solve the second order constant coefficient differential equation $y'' - y' - 6y = 0$ with initial conditions $y(0) = 1$ and $y'(0) = 2$.

$2y'' - 9y' - 5 = 0$

Solve the differential equation $y'' - 4y' + 4y = 0$.

Solve the differential equation $3y'' + 2y' + y = 0$.

Solve the initial value problem $y'' - y' - 12y = 0$ with the initial conditions $y(0) = 0$ and $y'(0) = 14$.

Solve the differential equation $\frac{du}{dt} = 2t + \frac{\sec^2 t}{2u}$ with the initial condition $u(0) = -5$.

Solve the differential equation $\frac{dy}{dx} - 3y = 0$ with the initial condition $y(0) = -1$.

Solve the differential equation $y' = \frac{x^2}{y}$ by showing that it is separable and using integration to find the solution.

Solve the initial value problem: $y' = 8xy + 3y$ with $y(0) = 5$.

What is the general solution to this differential equation: $\frac{\partial U}{\partial t} = \alpha \frac{\partial^2 U}{\partial x^2}$ with boundary conditions $U(0, t) = 0$ and $U(L, t) = 0$?

Given a circular rod where the temperature at one end is the same as the temperature at the other end ($U(-L, T) = U(L, T)$) and the heat flow at both ends is also equal, solve the heat equation with the initial conditions provided.

Solve the heat equation for a one-dimensional iron rod with prescribed temperature of 0 Celsius at both ends and a known starting temperature across the rod using the separation of variables technique.

Solve the 1-D heat equation for a rod with length $L$, where the ends of the rod are held at a fixed temperature of zero, starting with an initial temperature $T_0$.

Sketch a slope field for $y' = 2xy$ at the indicated points and sketch a solution that passes through (1, 1).

Consider the differential equation $\frac{dy}{dx} = \frac{1}{2}x + y - 1$. Part (a) on the axis provided sketch a slope field for the given differential equation at the nine points indicated.

Find the second derivative in terms of $x$ and $y$. Describe the region in the $xy$-plane in which all solution curves to the differential equation are concave up.

Let $y = f(x)$ be a particular solution to the differential equation with the initial condition $f(0) = 1$. Does $f$ have a relative minimum, a relative maximum or neither at $x = 0$? Justify your answer.

Find the values of the constants $m$ and $b$ for which $y = mx + b$ is a solution to the differential equation $\frac{dy}{dx} = \frac{1}{2}x + y - 1$.

If I'm looking at the point (1, -2), write the equation for the line tangent to the function that satisfies the differential equation through this point.

If I give you the slope field for this differential equation and you want to find a particular solution that goes through the point (0, 2), sketch the solution by following the arrows in the slope field.