I. Applications of separeble differential equations. 2.5: 1, 4, 5, 8, 17, 21.

II. Solve the following linear differential equations. Find the general solution if no initial condition is present. These problem are based on Section 2.1

1. xy' + 2y = sin(x); assume x>0. (Why assume this??)

2. xy' + y = e^x; assume x>0.

3. y' + y/(x-1) = 1/(x-3) ; y(2)=1.

4. y' + y/(x-1) = 1/(x-3) ; y(0)=1.

5. x²y' -3y = 2

6. y' + 2y = te^-2t; y(1)=0.

7. y' + 2y/t = cos(t)/t²; y(pi)=0, t>0.

8. t³y' + 4t²y = e^-t; y(-1)=0.

9. y' - 2y = e^2t

10. y' -2y = t²e^2t

III. Computer Problem: use either Maple or Matlab. This is based on Problem 26 in Section 2.1

Consider y' + 2y/3 = 1 -t/2, with y(0) = y₀. Sovle this for y₀ = -2, -1, 0, 1, and 2. Plot these and the direction field on one grid. Estimate the value of y₀ for which the solution touches, but does not cross, the t-axis. Plot this solution as well.

Now, use your brain to solve for the exact value of y₀ for this case.

IV. Theory Problem: (Similar to #30 in 2.1).

Consider y' + ay = e^bt. What conditions on a & b will guarantee that the limit of y(t) goes to 0 as t goes to infinity? Hint: be careful when a = -b.