Module 4: Practice Problems
Instructions: Work through each problem on paper first, then reveal the solution. These 10 problems cover homogeneous equations, all three characteristic equation cases, undetermined coefficients, and variation of parameters.
Problem 1: Distinct Real Roots
Solve y'' - 7y' + 12y = 0.
Solution
r² - 7r + 12 = (r-3)(r-4) = 0. y = c1e3t + c2e4t.
Problem 2: Repeated Root
Solve y'' + 10y' + 25y = 0.
Solution
r² + 10r + 25 = (r+5)² = 0. r = -5 repeated. y = (c1 + c2t)e-5t.
Problem 3: Complex Roots
Solve y'' - 6y' + 25y = 0.
Solution
r = (6 ± √(36-100))/2 = (6 ± 8i)/2 = 3 ± 4i. y = e3t(c1cos 4t + c2sin 4t).
Problem 4: IVP with Complex Roots
Solve y'' + 4y = 0, y(0) = 3, y'(0) = -2.
Solution
r = ±2i. y = c1cos 2t + c2sin 2t. y(0)=3: c1=3. y'=-2c1sin 2t + 2c2cos 2t. y'(0)=-2: 2c2=-2, c2=-1. y = 3cos 2t - sin 2t.
Problem 5: Undetermined Coefficients (Polynomial)
Solve y'' + 3y' + 2y = 6.
Solution
yh: (r+1)(r+2)=0, yh = c1e-t + c2e-2t. Guess yp = A. 0+0+2A=6, A=3. y = c1e-t + c2e-2t + 3.
Problem 6: Undetermined Coefficients (Exponential)
Solve y'' - 4y = 5e3t.
Solution
yh: r²-4=0, r=±2. yh=c1e2t+c2e-2t. Guess yp=Ae3t. 9A-4A=5, A=1. y=c1e2t+c2e-2t+e3t.
Problem 7: Undetermined Coefficients (Trig)
Solve y'' + 9y = cos 2t.
Solution
yh=c1cos 3t+c2sin 3t. Guess yp=Acos 2t+Bsin 2t. -4A+9A=1, 5A=1, A=1/5. 5B=0, B=0. yp=(1/5)cos 2t.
Problem 8: Modification Rule
Solve y'' - 4y' + 4y = e2t.
Solution
r=2 repeated. yh=(c1+c2t)e2t. e2t and te2t both in yh, so guess yp=At²e2t. Substituting gives 2A=1, A=1/2. yp=(1/2)t²e2t.
Problem 9: Variation of Parameters
Find yp for y'' + y = sec(t) using variation of parameters.
Solution
y1=cos t, y2=sin t, W=1. yp=-cos t ∫sin t sec t dt + sin t ∫cos t sec t dt = -cos t(-ln|cos t|) + sin t(t) = cos t ln|cos t| + t sin t.
Problem 10: Wronskian
Compute the Wronskian of y1 = e3t and y2 = e-t.
Solution
W = e3t(-e-t) - e-t(3e3t) = -e2t - 3e2t = -4e2t ≠ 0 (linearly independent).