Question 1

Given the functions $$f_1(x$$) = $$e^{2x}$$, $$f_2(x$$) = $$e^{-2x}$$, and $$f_3(x) = 1$, use the Wronskian to determine whether these functions are linearly independent on $$(-\infty, \infty)$$. Which of the following is correct?

Accepted Answer

They are linearly independent because the Wronskian is nonzero for all $$x.$$

Question 2

Consider the second-order linear homogeneous ODE $$y'' - 4y = 0. $$Which of the following is the general solution?

Accepted Answer

$$y(x) = C_1$ e$$^{2x}$$ + C_2$ e$$^{-2x}$$

Question 3

A mass-spring system is modeled by $$m x'' + c x' + k x = 0. If$$ $$c^2 - 4mk$$ \u003c 0$$, what type of damping does the system exhibit?

Accepted Answer

Underdamped

Question 4

Which of the following is Euler's formula?

Accepted Answer

$$e^{ix}$$ = \cos x + i \sin x$$

Question 5

Given the ODE $$y'' + 2y' + y = 0$$, what is the nature of its roots and the general solution?

Accepted Answer

Repeated real root; $$y(x) = (C_1$ + C_2$ x) e$$^{-x}$$

Question 6

For the nonhomogeneous ODE $$y'' + y = \sin x$$, which method is most appropriate to find a particular solution?

Accepted Answer

Undetermined Coefficients

Question 7

Suppose you have the ODE $$y'' + 4y = 0$$ with initial conditions $$y(0) = 2$$, $$y'(0) = 0. $$What is the solution?

Accepted Answer

$$y(x) = 2 \cos(2x)$$

Question 8

Given the ODE $$y'' - 2y' + 2y = 0$$, what is the general solution?

Accepted Answer

$$y(x) = e$$^{x}$$(C_1$ \cos x + C_2$ \sin x)$$

Question 9

A circuit with resistance $$R$$, inductance $$L$$, and capacitance $$C is $$modeled by $$LQ'' + RQ' + \frac{1}{C}Q = 0. If$$ $$R^2 = 4L/C$, what is the behavior of the circuit?

Accepted Answer

Critically damped (no oscillation, fastest return to equilibrium)

Question 10

Which of the following best describes the method of Variation of Parameters for solving $$y'' + p(x)y' + q(x)y = f(x)$$?

Accepted Answer

Use the complementary solution and integrate to find a particular solution

Question 11

For the ODE $$y'' + 9y = 0$$, which of the following is a fundamental set of solutions?

Accepted Answer

$$\{\cos 3x, \sin 3x\}$$

Question 12

A mass-spring-damper system is modeled by $$m x'' + c x' + k x = 0. If$$ $$m = 1 kg$$, $$c = 6 Ns/m$$, and $$k = 9 N/m$$, what is the nature of the system's response?

Accepted Answer

Overdamped

Question 13

Given the ODE $$y'' + 2y' + 5y = 0$$, what is the general solution?

Accepted Answer

$$y(x) = e$$^{-x}$$(C_1$ \cos 2x + C_2$ \sin 2x)$$

Question 14

Which of the following is the correct formula for the Wronskian of two functions $$y_1(x$$)$$ and y_2$(x)$$?

Accepted Answer

$$W(y_1$, y_2$) = y_1$ y_2$' - y_1$' y_2$

Question 15

For the ODE $$y'' + 4y = \cos 2x$$, what is a particular solution?

Accepted Answer

$$y_p(x) = \frac{1}{8} x \sin 2x$$

Question 16

If the characteristic equation of a second-order ODE has complex roots $$r = \alpha \pm i \beta$$, what is the general solution?

Accepted Answer

$$y(x) = e$$^{\alpha x}$$(C_1$ \cos \beta x + C_2$ \sin \beta x)$$

Question 17

A car's suspension system can be modeled as a mass-spring-damper system. If the system is critically damped, what is the expected behavior after hitting a bump?

Accepted Answer

The car returns to equilibrium as quickly as possible without oscillating.

Question 18

Which of the following is the correct formula for the complementary solution $$y_c of a $$second-order linear ODE with constant coefficients?

Accepted Answer

$$y_c = C_1$ y_1$(x) + C_2$ y_2$(x)$$

Question 19

In the method of undetermined coefficients, what is the first step when solving $$y'' + y = e$$^{x}$$?

Accepted Answer

Guess a particular solution of the form $$A e$$^{x}$$

Study Notes: Differential Equations and Linear ODEs (Calculus II/III)

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