Differential Equations

Given a circular region, with the top half at a temperature of 1 and the bottom half at a temperature of -1, find the temperature distribution inside the circle that reaches equilibrium.

Solve the Laplace equation for a heated plate with the following boundary conditions: The temperature is zero on the left and right edges, the bottom edge is insulated (derivative is zero), and the top edge has a given function describing the temperature distribution.

Solve the Laplace equation for a region where non-zero functions exist on all four boundaries, using the concept of superposition to combine the solutions from two separate problems, each addressing different boundary conditions.

Find the solution for the Laplace equation $\Delta^2 u = 0$ in two dimensions with the given boundary conditions.

Solve the differential equation using Laplace transforms: $y'' + 5y' + 6y = 0$ with initial conditions $y(0) = 0$ and $y'(0) = 1$.

Consider the differential equation $y'' - 2y' + 5y = 0$ with initial conditions $y(0) = 2$ and $y'(0) = -4$. Solve the initial value problem using the Laplace transform.

Solve the differential equation $y' + 4y = e^{4t}$ using Laplace transforms, given the initial condition $y(0) = 3$.

Solve a second order non-homogeneous ordinary differential equation (ODE) using the Laplace Transform with given initial conditions.

Find the Laplace transform of a piecewise function using unit step functions.

Solve the differential equation using the Laplace transform, considering the unit step function with initial conditions provided.

Given a piecewise function $f(t) = t$ for $0 \, \leq \, t \, < \, a$ and $0$ otherwise, express this function using step functions and compute its Laplace Transform.

Derive the equation of motion for a mass-spring-damper system using Newton's second law.

Using a Laplace Transform, solve the second-order linear differential equation of motion for a damped harmonic oscillator and rearrange it for a transfer function.

A mass $m$ is attached to a spring on a frictionless surface. Initially at rest at an equilibrium position. When displaced and released, determine the differential equation describing its motion using Newton's Second Law and Hooke's Law. Solve the differential equation for the position of the mass over time.

A typical vibrating mechanical system consisting of mass, spring and a viscous damper. This is an over-damped case where the roots of the characteristic equation are real and not repeated.

Given a mechanical system with a mass, spring, and damper, disturbed by initial displacement with no initial velocity, derive and solve the differential equation: $m x'' + c x' + k x = 0$ with given values: $m = 1 \text{ kg}$, $c = 3 \text{ Ns/m}$, $k = 2 \text{ N/m}$, and initial conditions $x(0) = 0.01 \text{ m}$, $x'(0) = 0$.

Solve the non-homogeneous ordinary differential equation: $y'' - 2y' - 3y = 3e^{2t}$ using the method of undetermined coefficients.

Solve the non-homogeneous differential equation: $y'' - 2y' - 3y = 3e^{-t}$ using the method of undetermined coefficients, accounting for overlapping with homogeneous solutions.

Using the method of undetermined coefficients, solve a non-homogeneous ordinary differential equation where the particular solution involves algebraic terms.

Find the form of the particular solution for the seventh-order non-homogeneous differential equation by solving the homogeneous case first.