Differential Equations

Solve the differential equation $\frac{dy}{dt} = ky$.

Given that a lake is affected by an algae bloom covering 1.5 square meters on October 3rd, and grows at a rate proportional to the quantity present covering 4 square meters on October 15th, use a differential equation to find an expression for the area covered at time T. How long will it take to completely cover a lake surface of 5 square kilometers if growth continues in this fashion?

Using the Wronskian method, determine if the functions $y_1 = 2t - 1$, $y_2 = t^2 + 5$, and $y_3 = 4t - 7$ are linearly independent.

For two functions $y_1 = e^{2x}$ and $y_2 = e^{-2x}$, determine if they are linearly independent using the Wronskian test.

Using the Wronskian test, determine if the functions $y_1 = \, \cos(x)$ and $y_2 = \, \sin(x)$ are linearly independent.

Using the Improved Euler's method, solve for the approximate value of $y$ at $x = 1.3$ for the differential equation $y' = x \cdot y$ with the initial condition $y(1) = 1$ and a step size $h = 0.1$.

Using the Euler and improved Euler techniques, approximate the values of $y$ given the initial conditions and a step size.

Given $\frac{dy}{dx} = f(x, y)$ with $x_0 = 3, y_0 = 2$, step size $h = 0.1$, approximate $y$ when $x = 3.1$ using the Euler method.

Using Euler's method with step size $h = 0.1$, approximate $y$ when $x = 1.1$ given that $x_0 = 1, y_0 = 1$ and $f(x, y) = x + 3 + \sin(y)$.

Using the improved Euler's method, approximate $y$ when $x = 1.2$ given that $x_0 = 1, y_0 = 1$, $h = 0.1$, and $f(x, y) = x + 3 + \, \sin(y)$.

Using the improved Euler method, with initial conditions $x_0 = 0$ and $y_0 = 1$, and a step size $h = 0.5$, calculate the value of $y$ for $x = 0.5$.

Use Runge-Kutta with step size $H = 0.1$ to estimate $y(0.2)$ in the initial value problem $y' = t^2 + y^2$ and $y(0) = 1$.

Compute the inverse Laplace transform of the expression $\frac{2}{(s+1)^3}$.

Find the inverse Laplace transform of $\frac{s}{s^2 - 2s + 5}$ using the method of completing the square.

Compute the inverse Laplace transform of a rational function using partial fractions when the denominator has repeated linear factors.

Find the inverse Laplace transform of $\frac{3s + 8}{s^2 + 2s + 5}$

Find the inverse Laplace transform of the function $f(s) = \frac{1}{s - 3} - \frac{16}{s^2 + 9}$.

Dirichlet problem for the Laplace equation with the region defined as the upper half plane.

Newman's problem for the Laplace equation in the upper half plane.

Robin's problem example: $u_{xx} + u_{yy} = 0$, in polar coordinates $\Delta^2 u = 0$ on a disk with radius 1. Boundary condition: $\frac{\partial u}{\partial r}$ at $r = 1$ plus $h$ into Function $f(x)$.