Practice Test 1 - Differential Equations

20 questions. Answer them in any order, then submit to see your score and every explanation.

A bacterial culture doubles in size every 4 hours. If the culture starts with 500 bacteria and grows according to the differential equation $\frac{dN}{dt} = kN$ , what is the value of the growth constant $k$ ?
For the nonlinear system $\frac{dx}{dt} = x(2-x-y)$ , $\frac{dy}{dt} = y(1-x-2y)$ , the equilibrium point $(1, \frac{1}{2})$ has been identified. To determine its stability, what is the trace of the Jacobian matrix evaluated at this equilibrium?
The population of an endangered species is modeled by a logistic equation of the form $\frac{dP}{dt} = kP(1 - P/500)$ . Initially, the population is $P(0) = 50$ . After 10 years, the population has grown to $P(10) = 100$ . What is the value of the growth constant $k$ ?
The function $F(s) = \frac{3s^2 + 2s - 1}{s^3 + s^2}$ represents the Laplace transform of some function $f(t)$ . Before applying partial fraction decomposition, what is the correct factorization of the denominator?
Consider the differential equation $y'' - 4y' + 4y = 8e^{2x} + 12x$ . When applying the method of undetermined coefficients, what form should the particular solution take?
Which of the following functions, defined by an integral, is a solution to the initial value problem $y' + 2xy = 1$ , with the initial condition $y(0)=0$ ?
The general solution to a differential equation is $y = C_1 x^2 + C_2 x + C_3$ . Two particular solutions are given: $y_1 = 2x^2 + 3x - 1$ and $y_2 = x^2 - x + 4$ . If $y_3 = ay_1 + by_2$ where $a$ and $b$ are constants with $a + b = 1$ , which statement about $y_3$ is most accurate?
The motion of a mechanical system is described by $3y'' + 15y = -12y'$ . Which of the following is the general solution $y(t)$ for the system's displacement?
The differential equation $(y \tan x) dx + dy = 0$ is made exact by an integrating factor of the form $\mu(x) = \sec^k x$ . What is the value of $k$ ?
A mass-spring system undergoes damped oscillations described by $x(t) = 0.2e^{-0.5t}\cos(3t + \frac{\pi}{4})$ meters. At what time does the mass first cross the equilibrium position ( $x = 0$ ) after $t = 0$ ?
A resistor-capacitor (RC) circuit with resistance $R = 2$ ohms and capacitance $C = 0.5$ farads is connected to a voltage source $V(t) = 10\sin(t)$ volts. The charge $q(t)$ on the capacitor satisfies $R\frac{dq}{dt} + \frac{q}{C} = V(t)$ . What is the steady-state charge amplitude?
The transfer function of a linear time-invariant system is given by $H(s) = \frac{1}{s^2+2s+k}$ . For which of the following values of the parameter $k$ will the impulse response $h(t) = \mathcal{L}^{-1}\{H(s)\}$ be a product of a decaying exponential and a sinusoidal function?
Consider the function $f(t) = 3\delta(t-2) + 5\delta(t-4)$ . If $g(t) = \int_{0}^{t} f(\tau) d\tau$ , what is the value of $g(5)$ ?
A researcher observes that in a real predator-prey system, the predator population peaks consistently lag behind prey population peaks by approximately one-quarter of the total cycle period. In the context of the Lotka-Volterra model $\frac{dx}{dt} = ax - bxy$ and $\frac{dy}{dt} = -cy + dxy$ , what does this phase relationship indicate about the system dynamics?
A nonlinear system has been linearized near the critical point $(2, -1)$ , yielding the Jacobian matrix $J = \begin{pmatrix} -3 & 4 \\ -2 & 1 \end{pmatrix}$ . The eigenvalues are $\lambda_1 = -1 + 2i$ and $\lambda_2 = -1 - 2i$ . Which classification is correct for the behavior near this critical point?
The growth of a bacterial culture is modeled by $dN/dt = N(0.8 - 0.0002N)$ , where $N$ is the number of bacteria and $t$ is time in hours. What is the limiting value of the relative growth rate, $(1/N)(dN/dt)$ , as $t \to \infty$ ? (Assume the initial population is positive and not equal to the carrying capacity.)
Determine the largest interval of existence for the solution to the initial value problem $y' - \frac{3}{2x}y = -\frac{1}{2}x y^3$ , with $y(1)=1$ .
Carbon-14 has a half-life of 5,730 years. An archaeological sample shows 25% of the original C-14 activity. If measurement uncertainty is ±200 years, what is the most likely age range of the sample?
Find the Laplace transform of the function $f(t) = \cos(2t)(u_{\pi/2}(t) - u_{3\pi/2}(t))$ .
Based on the given data and carrying capacity estimate, what is the most significant concern about the validity of the logistic model for this population?

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