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4. y''(t) + 2y'(t) = 8t with y(0) = y'(0) = 0. 5. y''(t) + y'(t) = g(t) where g(t) = {4, 0 ? t ? 2; t + 2, 2 ? t} and y(0) = y'(0) = 0. 6. y'''(t) - 6y''(t) + 11y'(t) - 6y(t) = 1 with y(0) = y'(0) = y''(0) = 0. Laplace transforms can be used to solve simultaneous differential equations. Solve the following sets of differential equations using Laplace transforms: 7. x'(t) + 2y'(t) - 2y(t) = t x(t) + y'(t) - y(t) = 1 with x(0) = y(0) = 0. 8. x'(t) - x(t) + 2y(t) = 0 3x(t) + y'(t) - 2y(t) = 0 with x(0) = 1 and y(0) = 2. 9. x'(t) - x(t) - 2y(t) = t -2x(t) + y'(t) - y(t) = t with x(0) = 2 and y(0) = 4. 10. Solve y''(t) + 2y'(t) + y(t) = g(t) where g(t) = {e^{-t}, 0 ? t < 1; 0, t ? 1}. 11. Solve x''(t) + ?_0^2 x(t) = F_0 ?(t - t_0) with x(0) = a and x'(0) = 0. 12. The differential equation governing an LC circuit is L d^2q/dt^2 + q/C = E(t). Solve this equation for q(t) and i(t) = dq/dt for the case E(t) = E_0 and q(0) = i(0) = 0. 13. Determine i(t) for an LR circuit where E(t) = E_0 and i(0) = 0. Interpret your result physically. 14. Solve the previous problem with E(t) = E_0 cos ?t. The quantity Z^2 = L^2 ?^2 + R^2 that arises naturally is called the impedance of this circuit. 15. Determine i(t) for an RC circuit with E(t) = E_0 sin ?t. Assume that q(0) = 0 and i(0) = 0. Express your answer in terms of Z^2 = R^2 + 1/?^2 C^2, which is the impedance for this circuit. 16. A beam with one end clamped and the other end free is called a cantilever beam. Determine the deflection of a cantilever beam that is subject to a constant load -w_0 over its entire length a. 17. Determine the deflection of the beam in the previous problem if it is clamped at both ends. 18. A beam of length 2a is clamped at both ends and has an applied load w_0 concentrated at its midpoint. Determine the deflection of the beam. Represent the load by -w_0 ?(x - a). 19. A cantilever beam (see Problem 16) of length 2a is subject to a load described by W(x) = { - (w_0 / a) (a - x), 0 ? x ? a 0, a < x ? 2a } Determine the deflection of the beam. 20. Re-do Problem 19 with the beam clamped at both ends. 21. Determine the deflection of a beam of length 2a that is simply supported at each end and with a uniform load -w_0.

          4. y''(t) + 2y'(t) = 8t with y(0) = y'(0) = 0.

5. y''(t) + y'(t) = g(t) where g(t) = {4, 0 ? t ? 2; t + 2, 2 ? t} and y(0) = y'(0) = 0.

6. y'''(t) - 6y''(t) + 11y'(t) - 6y(t) = 1 with y(0) = y'(0) = y''(0) = 0.

Laplace transforms can be used to solve simultaneous differential equations. Solve the following sets of differential equations using Laplace transforms:

7. x'(t) + 2y'(t) - 2y(t) = t
   x(t) + y'(t) - y(t) = 1
   with x(0) = y(0) = 0.

8. x'(t) - x(t) + 2y(t) = 0
   3x(t) + y'(t) - 2y(t) = 0
   with x(0) = 1 and y(0) = 2.

9. x'(t) - x(t) - 2y(t) = t
   -2x(t) + y'(t) - y(t) = t
   with x(0) = 2 and y(0) = 4.

10. Solve y''(t) + 2y'(t) + y(t) = g(t) where g(t) = {e^{-t}, 0 ? t < 1; 0, t ? 1}.

11. Solve x''(t) + ?_0^2 x(t) = F_0 ?(t - t_0) with x(0) = a and x'(0) = 0.

12. The differential equation governing an LC circuit is L d^2q/dt^2 + q/C = E(t). Solve this equation for q(t) and i(t) = dq/dt for the case E(t) = E_0 and q(0) = i(0) = 0.

13. Determine i(t) for an LR circuit where E(t) = E_0 and i(0) = 0. Interpret your result physically.

14. Solve the previous problem with E(t) = E_0 cos ?t. The quantity Z^2 = L^2 ?^2 + R^2 that arises naturally is called the impedance of this circuit.

15. Determine i(t) for an RC circuit with E(t) = E_0 sin ?t. Assume that q(0) = 0 and i(0) = 0. Express your answer in terms of Z^2 = R^2 + 1/?^2 C^2, which is the impedance for this circuit.

16. A beam with one end clamped and the other end free is called a cantilever beam. Determine the deflection of a cantilever beam that is subject to a constant load -w_0 over its entire length a.

17. Determine the deflection of the beam in the previous problem if it is clamped at both ends.

18. A beam of length 2a is clamped at both ends and has an applied load w_0 concentrated at its midpoint. Determine the deflection of the beam. Represent the load by -w_0 ?(x - a).

19. A cantilever beam (see Problem 16) of length 2a is subject to a load described by

   W(x) = { - (w_0 / a) (a - x),  0 ? x ? a
            0,                    a < x ? 2a }

   Determine the deflection of the beam.

20. Re-do Problem 19 with the beam clamped at both ends.

21. Determine the deflection of a beam of length 2a that is simply supported at each end and with a uniform load -w_0.

4. y”(t) + 2y'(t) = 8t with y(0) = y'(0) = 0.

5. y”(t) + y'(t) = g(t) where g(t) = 4, 0 ? t ? 2; t + 2, 2 ? t and y(0) = y'(0) = 0.

6. y”'(t) - 6y”(t) + 11y'(t) - 6y(t) = 1 with y(0) = y'(0) = y”(0) = 0.

Laplace transforms can be used to solve simultaneous differential equations. Solve the following sets of differential equations using Laplace transforms:

7. x'(t) + 2y'(t) - 2y(t) = t
x(t) + y'(t) - y(t) = 1
with x(0) = y(0) = 0.

8. x'(t) - x(t) + 2y(t) = 0
3x(t) + y'(t) - 2y(t) = 0
with x(0) = 1 and y(0) = 2.

9. x'(t) - x(t) - 2y(t) = t
-2x(t) + y'(t) - y(t) = t
with x(0) = 2 and y(0) = 4.

10. Solve y”(t) + 2y'(t) + y(t) = g(t) where g(t) = e^-t, 0 ? t < 1; 0, t ? 1.

11. Solve x”(t) + ?0^2 x(t) = F0 ?(t - t0) with x(0) = a and x'(0) = 0.

12. The differential equation governing an LC circuit is L d^2q/dt^2 + q/C = E(t). Solve this equation for q(t) and i(t) = dq/dt for the case E(t) = E0 and q(0) = i(0) = 0.

13. Determine i(t) for an LR circuit where E(t) = E0 and i(0) = 0. Interpret your result physically.

14. Solve the previous problem with E(t) = E0 cos ?t. The quantity Z^2 = L^2 ?^2 + R^2 that arises naturally is called the impedance of this circuit.

15. Determine i(t) for an RC circuit with E(t) = E0 sin ?t. Assume that q(0) = 0 and i(0) = 0. Express your answer in terms of Z^2 = R^2 + 1/?^2 C^2, which is the impedance for this circuit.

16. A beam with one end clamped and the other end free is called a cantilever beam. Determine the deflection of a cantilever beam that is subject to a constant load -w0 over its entire length a.

17. Determine the deflection of the beam in the previous problem if it is clamped at both ends.

18. A beam of length 2a is clamped at both ends and has an applied load w0 concentrated at its midpoint. Determine the deflection of the beam. Represent the load by -w0 ?(x - a).

19. A cantilever beam (see Problem 16) of length 2a is subject to a load described by

W(x) = - (w0 / a) (a - x), 0 ? x ? a
0, a < x ? 2a

Determine the deflection of the beam.

20. Re-do Problem 19 with the beam clamped at both ends.

21. Determine the deflection of a beam of length 2a that is simply supported at each end and with a uniform load -w0.

Added by Ashley W.

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