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# 17R: Chapter 17 Review Exercises

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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.

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 $$y″+2y′+10=0.$$

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. $$y″−2y=0$$

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

6. $$y''−3y+2y= \cos (t)$$

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

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

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

10. $$y''+2y′+y=0$$

11. $$y''−2y′+10y=4x$$

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

12. $$y''= \cos (x)+2y′+y$$

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

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

14. $$y''=3y′+xe^{−x}$$

15. $$y''−x^2=−3y′−\frac{9}{4}y+3x$$

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

16. $$y''=2 \cos x+y′−y$$

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

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

$$y=e^{−2x} \sin (\sqrt{2}x)$$

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

$$y=\dfrac{e^{1−x}}{e^4−1}(e^{4x}−1)$$

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.

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

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.)

$$b=\sqrt{a}$$