You’ve made it through the entire Differential Equations course. This review spans everything: direction fields, separable and exact equations, growth and decay models, mixing problems, second-order equations (homogeneous and nonhomogeneous), Laplace transforms, systems, phase plane analysis, numerical methods, and spring-mass applications.
That’s a lot of ground. You’ve gone from “what even is a differential equation?” to modeling real physical systems with multiple interacting components. If you can pass this review, you genuinely understand the subject.
Aim for 15 out of 20 to pass. Take your time. Sketch a direction field or write out the characteristic equation if you’re stuck on a question. The techniques are all connected, and sometimes the best way to remember a formula is to re-derive it quickly from first principles.
A differential equation is called separable if it can be written as: A. $\frac{dy}{dx} = f(x) + g(y)$ B. $\frac{dy}{dx} = f(x) \cdot g(y)$ C. $y'' = f(x,y)$ D. $\frac{dy}{dx} = ky^2 + x$
The integrating factor for $\frac{dy}{dx} + P(x)y = Q(x)$ is: A. $e^{P(x)}$ B. $\int Q(x)\,dx$ C. $e^{\int P(x)\,dx}$ D. $\frac{1}{P(x)}$
An equation $M\,dx + N\,dy = 0$ is exact when: A. $M = N$ B. $M + N = 0$ C. $\frac{\partial M}{\partial x} = N$ D. $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$
The solution to $\frac{dP}{dt} = kP$ is: A. $P = kt + C$ B. $P = \frac{P_0}{1 - kt}$ C. $P = P_0 e^{kt}$ D. $P = k\ln t + C$
Newton's Law of Cooling is modeled by: A. $\frac{dT}{dt} = kT$ B. $\frac{dT}{dt} = k(T - T_a)^2$ C. $T'' + kT = 0$ D. $\frac{dT}{dt} = -k(T - T_a)$
For the characteristic equation $r^2 + 6r + 13 = 0$, the roots are: A. Real and distinct B. Repeated real C. Purely imaginary D. Complex (with nonzero real part)
The general solution for complex roots $\alpha \pm \beta i$ is: A. $C_1 e^{r_1 x} + C_2 e^{r_2 x}$ B. $(C_1 + C_2 x)e^{rx}$ C. $C_1\cos rx + C_2\sin rx$ D. $e^{\alpha x}(C_1\cos\beta x + C_2\sin\beta x)$
In undetermined coefficients, we multiply the guess by $x$ when: A. The right-hand side is zero B. The equation is first-order C. We have complex roots D. The guess overlaps with the homogeneous solution
Variation of parameters is more general because: A. It only works for polynomials B. It requires no integration C. It works for almost any continuous $g(x)$ D. It only solves homogeneous equations
$\mathcal{L}\{f'(t)\}$ equals: A. $sF(s)$ B. $s^2 F(s)$ C. $F(s)/s$ D. $sF(s) - f(0)$
A second-order ODE is converted to a system by setting: A. $x_1 = y''$ B. $x_1 = p$ C. $x_1 = y$, $x_2 = y'$ D. $x_1 = g(t)$
In a phase portrait, a saddle point occurs when eigenvalues: A. Are both negative B. Are complex C. Are repeated D. Have opposite signs
Euler's method uses the update rule: A. $y_{n+1} = y_n + hf(x_n, y_n)$ B. $y_{n+1} = y_n + f(x_{n+1}, y_n)$ C. $y_{n+1} = hf(x_n, y_n)$ D. $y = y_0 e^{kx}$
The spring-mass equation is: A. $\frac{dP}{dt} = rP(1 - P/K)$ B. $\frac{dT}{dt} = -k(T - T_a)$ C. $mx'' + cx' + kx = F(t)$ D. $y'' + y = \sin t$
Critically damped motion occurs when: A. $c = 0$ B. $c^2 \gt 4mk$ C. $c^2 \lt 4mk$ D. $c^2 = 4mk$
Laplace transforms are especially useful for: A. Homogeneous equations only B. Nonlinear equations C. IVPs with discontinuous inputs D. Partial differential equations
In a direction field, horizontal line segments indicate: A. Rapid growth B. Vertical asymptotes C. Equilibrium solutions (where $dy/dx = 0$) D. Oscillatory behavior
For the logistic equation, as $t \to \infty$ the population approaches: A. Zero B. Infinity C. The initial population D. The carrying capacity $K$
The Wronskian is used in: A. Euler's method B. Undetermined coefficients C. Direction fields D. Variation of parameters
Which tool handles irregular forcing functions best? A. Undetermined coefficients B. Direction fields C. Variation of parameters D. Euler's method
Retry Quiz Retrying will remove your ✅ checkmark until you pass again.
