Math 307G Midterm 2

Do all problems. Use no calculators.

When solving differential equations, solve by a method different from the power series method if possible. If you do solve by the power series method, compute the first three non-zero coefficients following the coefficients a0 and a1 of the series for y.

  1. (20 points) For this differential equation, solve the initial value problem y(0) = 0, y'(0) = 1:

    2. y'' + y' – 2y = 2t

     

  2. (20 points) Find the general solution of

    3. y'' + 4y = 8 sin 2t

  3. (20 points)

    4. (a) Find the general solution of

    y'' – (1 + x)y' - 6y = 0

     

    (b) Solve the initial value problem y(0) = 2, y'(0) = 0.

     

  4. (20 points)
  1. Compute the second Taylor polynomial a0 + a1x + a2 x2 for the function

f(x) = (1+x)1/2.

 

(b) Compute the remainder (error) term for this Taylor polynomial.

 

(c) Approximate f(0.3) using this Taylor polynomial. Then use the error term to determine the accuracy of the Taylor approximation.
  1. (10 points) Let y(x) = -3 cos 2x + 3 sin 2x.

Then y(x) can be written as R cos (2x – x0).
  1. Compute the amplitude R.
  2. Compute x0. (Note: You should be able to figure out x0 exactly, but if not, you will get most of the credit for specifying the angle x0 using inverse trig formulas provided you also specify exactly which quadrant x0 is in.)

 

  1. (10 points) DO NOT SOLVE the equation below, but tell the best choice of y that will give a solution by the method of undetermined coefficients.

y'' – 2y' + 1 = t et cos 2t