Applied Mathematics I (500.303)

Suggested Reading: Chapters 6 and 7 (especially if you've forgotten eigenvalues and eigenvectors.)

1. (Section 5.7)    (a) Show that the system of differential equations for the currents i₁(t) and i₂(t) shown in the figure below is
     
   (b) Use Laplace transforms to solve the system in part (a) if E = 60 volts, L = 1 henry, R = 50 ohms, C = 0.0001 farads and i₁ and i₂ are initially zero.
   (c) Determine the current i₃(t).

2. (Section 5.7)   A double pendulum oscillates in a vertical plane under the influence of gravity (see the figure below).   For small displacements , it can be shown that the differential equations of motion are
     
   Use Laplace tansforms to solve the system when m₁ = 3, m₂ = 1, l₁ = l₂ = 16, g = 32 ft/s²,

3. Kreyszig problem 7.2.6
4. Kreyszig problem 7.2.12
5. Kreyszig problem 7.5.13
6. Kreyszig problem 7.5.17

7. (Section 3.3)    Using the information given in the figure below, derive (do NOT solve) the system of differential equations describing the number of pounds of salt x_A, x_B, and x_C at any time in tanks A, B, and C, respectively.   (Note: The arrows on the diagram below give the direction in which the fluid is flowing.)
   
   
   Extra Credit: Solve the system above subject to x_A(0) = 50, x_B(0) = 0, x_C(0) = 25.
   

8. Solve the following system of differential equations.
   

   x1'(t) = 2x1 + x2 + x3	x1(0) = 0
   x2'(t) = 2x1 + 3x2 + 2x3	x2(0) = 1
   x3'(t) = x1 + x2 + 2x3	x3(0) = 2