Oscillations — Practice Problems and Worked Solutions

17 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Solve the undamped oscillator initial-value problem.

   $$y^{\prime \prime }+1y=0,y(0)=1,y^{\prime }(0)=0$$

   Answer: $\cos \left(x\right)$

   Step-by-step solution

   1. $$r=\pm 1i$$
   2. $$y=A\cos (1x)+B\sin (1x)$$
   3. $$y=\cos \left(x\right)$$
2. 2.

   Solve the undamped oscillator initial-value problem.

   $$y^{\prime \prime }+4y=0,y(0)=0,y^{\prime }(0)=1$$

   Answer: $\frac{\sin \left(2x\right)}{2}$

   Step-by-step solution

   1. $$r=\pm 2i$$
   2. $$y=A\cos (2x)+B\sin (2x)$$
   3. $$y=\frac{\sin \left(2x\right)}{2}$$
3. 3.

   Solve the undamped oscillator initial-value problem.

   $$y^{\prime \prime }+9y=0,y(0)=2,y^{\prime }(0)=-1$$

   Answer: $-\frac{\sin \left(3x\right)}{3}+2\cos \left(3x\right)$

   Step-by-step solution

   1. $$r=\pm 3i$$
   2. $$y=A\cos (3x)+B\sin (3x)$$
   3. $$y=-\frac{\sin \left(3x\right)}{3}+2\cos \left(3x\right)$$
4. 4.

   Solve the undamped oscillator initial-value problem.

   $$y^{\prime \prime }+1y=0,y(0)=-2,y^{\prime }(0)=2$$

   Answer: $2\sin \left(x\right)-2\cos \left(x\right)$

   Step-by-step solution

   1. $$r=\pm 1i$$
   2. $$y=A\cos (1x)+B\sin (1x)$$
   3. $$y=2\sin \left(x\right)-2\cos \left(x\right)$$
5. 5.

   Solve the undamped oscillator initial-value problem.

   $$y^{\prime \prime }+16y=0,y(0)=1,y^{\prime }(0)=1$$

   Answer: $\frac{\sin \left(4x\right)}{4}+\cos \left(4x\right)$

   Step-by-step solution

   1. $$r=\pm 4i$$
   2. $$y=A\cos (4x)+B\sin (4x)$$
   3. $$y=\frac{\sin \left(4x\right)}{4}+\cos \left(4x\right)$$
6. 6.

   Solve the undamped oscillator initial-value problem.

   $$y^{\prime \prime }+4y=0,y(0)=-1,y^{\prime }(0)=3$$

   Answer: $\frac{3\sin \left(2x\right)}{2}-\cos \left(2x\right)$

   Step-by-step solution

   1. $$r=\pm 2i$$
   2. $$y=A\cos (2x)+B\sin (2x)$$
   3. $$y=\frac{3\sin \left(2x\right)}{2}-\cos \left(2x\right)$$
7. 7.

   Solve the undamped oscillator initial-value problem.

   $$y^{\prime \prime }+1y=0,y(0)=3,y^{\prime }(0)=0$$

   Answer: $3\cos \left(x\right)$

   Step-by-step solution

   1. $$r=\pm 1i$$
   2. $$y=A\cos (1x)+B\sin (1x)$$
   3. $$y=3\cos \left(x\right)$$
8. 8.

   Solve the undamped oscillator initial-value problem.

   $$y^{\prime \prime }+4y=0,y(0)=2,y^{\prime }(0)=4$$

   Answer: $2\sin \left(2x\right)+2\cos \left(2x\right)$

   Step-by-step solution

   1. $$r=\pm 2i$$
   2. $$y=A\cos (2x)+B\sin (2x)$$
   3. $$y=2\sin \left(2x\right)+2\cos \left(2x\right)$$
9. 9.

   Solve the undamped oscillator initial-value problem.

   $$y^{\prime \prime }+9y=0,y(0)=1,y^{\prime }(0)=3$$

   Answer: $\sin \left(3x\right)+\cos \left(3x\right)$

   Step-by-step solution

   1. $$r=\pm 3i$$
   2. $$y=A\cos (3x)+B\sin (3x)$$
   3. $$y=\sin \left(3x\right)+\cos \left(3x\right)$$
10. 10.

    Solve the undamped oscillator initial-value problem.

    $$y^{\prime \prime }+1y=0,y(0)=0,y^{\prime }(0)=5$$

    Answer: $5\sin \left(x\right)$

    Step-by-step solution

    1. $$r=\pm 1i$$
    2. $$y=A\cos (1x)+B\sin (1x)$$
    3. $$y=5\sin \left(x\right)$$
11. 11.

    Solve the undamped oscillator initial-value problem.

    $$y^{\prime \prime }+16y=0,y(0)=2,y^{\prime }(0)=-4$$

    Answer: $-\sin \left(4x\right)+2\cos \left(4x\right)$

    Step-by-step solution

    1. $$r=\pm 4i$$
    2. $$y=A\cos (4x)+B\sin (4x)$$
    3. $$y=-\sin \left(4x\right)+2\cos \left(4x\right)$$
12. 12.

    Solve the undamped oscillator initial-value problem.

    $$y^{\prime \prime }+25y=0,y(0)=1,y^{\prime }(0)=0$$

    Answer: $\cos \left(5x\right)$

    Step-by-step solution

    1. $$r=\pm 5i$$
    2. $$y=A\cos (5x)+B\sin (5x)$$
    3. $$y=\cos \left(5x\right)$$
13. 13.

    Solve the undamped oscillator initial-value problem.

    $$y^{\prime \prime }+4y=0,y(0)=3,y^{\prime }(0)=2$$

    Answer: $\sin \left(2x\right)+3\cos \left(2x\right)$

    Step-by-step solution

    1. $$r=\pm 2i$$
    2. $$y=A\cos (2x)+B\sin (2x)$$
    3. $$y=\sin \left(2x\right)+3\cos \left(2x\right)$$
14. 14.

    Solve the undamped oscillator initial-value problem.

    $$y^{\prime \prime }+9y=0,y(0)=-1,y^{\prime }(0)=6$$

    Answer: $2\sin \left(3x\right)-\cos \left(3x\right)$

    Step-by-step solution

    1. $$r=\pm 3i$$
    2. $$y=A\cos (3x)+B\sin (3x)$$
    3. $$y=2\sin \left(3x\right)-\cos \left(3x\right)$$
15. 15.

    Solve the undamped oscillator initial-value problem.

    $$y^{\prime \prime }+1y=0,y(0)=4,y^{\prime }(0)=-1$$

    Answer: $-\sin \left(x\right)+4\cos \left(x\right)$

    Step-by-step solution

    1. $$r=\pm 1i$$
    2. $$y=A\cos (1x)+B\sin (1x)$$
    3. $$y=-\sin \left(x\right)+4\cos \left(x\right)$$
16. 16.

    Solve the undamped oscillator initial-value problem.

    $$y^{\prime \prime }+36y=0,y(0)=1,y^{\prime }(0)=6$$

    Answer: $\sin \left(6x\right)+\cos \left(6x\right)$

    Step-by-step solution

    1. $$r=\pm 6i$$
    2. $$y=A\cos (6x)+B\sin (6x)$$
    3. $$y=\sin \left(6x\right)+\cos \left(6x\right)$$
17. 17.

    Solve the undamped oscillator initial-value problem.

    $$y^{\prime \prime }+4y=0,y(0)=5,y^{\prime }(0)=-2$$

    Answer: $-\sin \left(2x\right)+5\cos \left(2x\right)$

    Step-by-step solution

    1. $$r=\pm 2i$$
    2. $$y=A\cos (2x)+B\sin (2x)$$
    3. $$y=-\sin \left(2x\right)+5\cos \left(2x\right)$$

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