17.R: Chapter 17 Review Exercises

True or False? Justify your answer with a proof or a counterexample.

1. If $y$ and $z$ are both solutions to $y''+2y′+y=0,$ then $y+z$ is also a solution.

Answer

True

2. The following system of algebraic equations has a unique solution:

$\begin{align*} 6z_1+3z_2 &=8 \\ 4z_1+2z_2 &=4. \end{align*}$

3. $y=e^x \cos (3x)+e^x \sin (2x)$ is a solution to the second-order differential equation $y″+2y′+10=0.$

Answer

False

4. To find the particular solution to a second-order differential equation, you need one initial condition.

In problems 5 - 8, classify the differential equations. Determine the order, whether it is linear and, if linear, whether the differential equation is homogeneous or nonhomogeneous. If the equation is second-order homogeneous and linear, find the characteristic equation.

5. $y″−2y=0$

Answer

second order, linear, homogeneous, $λ^2−2=0$

6. $y''−3y+2y= \cos (t)$

7. $\left(\dfrac{dy}{dt}\right)^2+yy′=1$

Answer

first order, nonlinear, nonhomogeneous

8. $\dfrac{d^2y}{dt^2}+t \dfrac{dy}{dt}+\sin^2 (t)y=e^t$

In problems 9 - 16, find the general solution.

9. $y''+9y=0$

Answer

$y=c_1 \sin (3x)+c_2 \cos (3x)$

10. $y''+2y′+y=0$

11. $y''−2y′+10y=4x$

Answer

$y=c_1e^x \sin (3x)+c_2e^x \cos (3x)+\frac{2}{5}x+\frac{2}{25}$

12. $y''= \cos (x)+2y′+y$

13. $y''+5y+y=x+e^{2x}$

Answer

$y=c_1e^{−x}+c_2e^{−4x}+\frac{x}{4}+\frac{e^{2x}}{18}−\frac{5}{16}$

14. $y''=3y′+xe^{−x}$

15. $y''−x^2=−3y′−\frac{9}{4}y+3x$

Answer

$y=c_1e^{(−3/2)x}+c_2xe^{(−3/2)x}+\frac{4}{9}x^2+\frac{4}{27}x−\frac{16}{27}$

16. $y''=2 \cos x+y′−y$

In problems 17 - 18, find the solution to the initial-value problem, if possible.

17. $y''+4y′+6y=0, \; y(0)=0, \; y′(0)=\sqrt{2}$

Answer

$y=e^{−2x} \sin (\sqrt{2}x)$

18. $y''=3y− \cos (x), \; y(0)=\frac{9}{4}, \; y′(0)=0$

In problems 19 - 20, find the solution to the boundary-value problem.

19. $4y′=−6y+2y″, \; y(0)=0, \; y(1)=1$

Answer

$y=\dfrac{e^{1−x}}{e^4−1}(e^{4x}−1)$

20. $y''=3x−y−y′, \; y(0)=−3, \; y(1)=0$

For the following problem, set up and solve the differential equation.

21. The motion of a swinging pendulum for small angles $θ$ can be approximated by $\dfrac{d^2θ}{dt^2}+\dfrac{g}{L}θ=0,$ where $θ$ is the angle the pendulum makes with respect to a vertical line, $g$ is the acceleration resulting from gravity, and $L$ is the length of the pendulum. Find the equation describing the angle of the pendulum at time $t,$ assuming an initial displacement of $θ_0$ and an initial velocity of zero.

Answer

$θ(t)=θ_0 \cos\left(\sqrt{\frac{g}{l}}t\right)$

In problems 22 - 23, consider the “beats” that occur when the forcing term of a differential equation causes “slow” and “fast” amplitudes. Consider the general differential equation $ay″+by= \cos (ωt)$ that governs undamped motion. Assume that $\sqrt{\frac{b}{a}}≠ω.$

22. Find the general solution to this equation (Hint: call $ω_0=\sqrt{b/a}$).

23. Assuming the system starts from rest, show that the particular solution can be written as$y=\dfrac{2}{a(ω_0^2−ω^2)} \sin \left(\dfrac{ω_0−ωt}{2}\right) \sin\left(\dfrac{ω_0+ωt}{2}\right).$

24. [T] Using your solutions derived earlier, plot the solution to the system $2y″+9y= \cos (2t)$ over the interval $t=[−50,50].$ Find, analytically, the period of the fast and slow amplitudes.

For the following problem, set up and solve the differential equations.

25. An opera singer is attempting to shatter a glass by singing a particular note. The vibrations of the glass can be modeled by $y″+ay= \cos (bt)$, where $y''+ay=0$ represents the natural frequency of the glass and the singer is forcing the vibrations at $\cos (bt)$. For what value $b$ would the singer be able to break that glass? (Note: in order for the glass to break, the oscillations would need to get higher and higher.)

Answer

$b=\sqrt{a}$