Differential Equations: Second Order Homogeneous Equations

Given a series RLC circuit with the equation $L \frac{d^2 i}{dt^2} + R \frac{di}{dt} + \frac{1}{C} i = 0$ , where the inductance $L = 0.2$ Henrys, resistance $R = 330$ Ohms, and capacitance $C = 20$ microfarads, and initial conditions $i(0) = 0$ and $\frac{di}{dt}(0) = 30$ , solve the second-order differential equation for the current $i$ as a function of time $t$

What if the characteristic equation does not have real roots, what if they are complex?

Solve a constant coefficient homogeneous linear 2nd order differential equation with complex roots of the form $r = -1 \pm i$. Derive the general solution form.

Find the particular solution for the differential equation $y'' + 4y' + 5y = 0$ with the initial conditions $y(0) = 1$ and $y'(0) = 0$.

What happens when you have two complex roots? Or essentially, when you're trying to solve the characteristic equation, when you're trying to solve that quadratic? The $B^2 - 4AC$ is negative, so you get the two roots end up being complex conjugates.

Let's say I had the differential equation $y'' + y' + y = 0$. What are the roots of the characteristic equation $r^2 + r + 1 = 0$ using the quadratic formula?

Given the second order differential equation $y y' = 6 (y')^2$, solve by making the substitution $y' = \frac{dp}{dx}$ and finding the general solution.

Find the general solution and a second solution of the differential equation $x^2 y'' + 5x y' - 5y = 0$ given that $y_1 = x$ is a known solution.

Find the general solution to the differential equation: $\frac{d^2y}{dx^2} + 4 \frac{dy}{dx} + 4y = 0$.

Derive the general form solution for a second-order constant coefficient linear homogeneous differential equation with repeated roots.

Solve the second-order linear constant coefficient homogeneous equation $y'' - 2y' + y = 0$ using the reduction of order method to find a second independent solution.

Find the general solution to the second order linear homogeneous differential equation $y'' + 2y' + y = 0$.

Given a second order linear homogeneous differential equation with a repeated root, find the general solution.

Solve the second order homogeneous linear differential equation $4y'' - 5y' - 6y = 0$ using the method of characteristic 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 second order linear homogeneous differential equation: $y'' + 2y' + 8y = 0$ using the quadratic formula to find the characteristic equation roots.