Differential 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 $t$

Using the systematic approach, find the initial and final values of the current through the inductor and the voltage across the capacitor for a second-order circuit before and after the switch is closed or a source is turned on.

Consider the differential equation: $\frac{dx}{dt} = x(1 - x)$. Identify the equilibrium points and determine their stability.

Come up with an example of an equation that exhibits an unstable equilibrium where one arrow is going into the equilibrium and one arrow is going out.

Is there any configuration of stable and unstable equilibria which could not occur as the phase-line for a differential equation?

Given an autonomous differential equation $\frac{dy}{dt} = f(y)$ where $f(y) = 1 + y (1 - y)$, identify the equilibrium solutions and determine their stability.

dy/dt = 3y(y - 2)

Given an autonomous differential equation $F(y) = 10 + 3y - y^2$, find the critical points and determine their stability as attractors or repellers by analyzing the sign of $F(y)$ in various regions divided by these critical points.

What is the maximum sustainable fishing-rate? At what rate do I know that I can fish without killing off the fish population entirely?

Analyze the behavior of the differential equation $\frac{dz}{dt} = 4z - z^3$.

For different values of parameter $c$, how does the behavior of the dynamical system described by $\frac{dz}{dt} = 4z - z^3 + c$ change? Summarize these changes in a bifurcation diagram.

Given a differential equation representing logistic population growth with harvesting: $\frac{dy}{dt} = y(K - y) - \alpha$, analyze how the changing harvesting rate $\alpha$ affects the population by using bifurcation diagrams. Identify bifurcation points and describe the stability of equilibrium solutions.

Consider the differential equation $\frac{dX}{dt} = xe^{-ax} - x^2$. Here, $a$ is the bifurcation parameter.

Sketch a bifurcation diagram and indicate the stability of each section created by the bifurcation diagram. Additionally, identify any thresholds where a change in long-term behavior occurs.

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?

Solve the differential equation: $\frac{dy}{dx} + \frac{2x}{1 + x^2} y = 4(1 + x^2)^2$ using an integrating factor.

Solve a non-exact differential equation using an integrating factor given the equation: $3x^2 y + x y^2 + x^3 + x^2 y$.