Module 4 Quiz
Test Your Knowledge
10 questions covering characteristic equations, all three root cases, undetermined coefficients, and variation of parameters.
1
Write the characteristic equation for 2y'' + 3y' - 5y = 0.
Answer: 2r² + 3r - 5 = 0. Using the quadratic formula: r = (-3 ± √(9+40))/4 = (-3 ± 7)/4. r = 1 or r = -5/2.
2
Solve y'' + 6y' + 9y = 0.
Answer: (r+3)² = 0, r = -3 repeated. y = (c1 + c2t)e-3t.
3
Solve y'' + 2y' + 10y = 0.
Answer: r = (-2 ± √(4-40))/2 = -1 ± 3i. y = e-t(c1cos 3t + c2sin 3t).
4
For y'' + 4y = 3t², what form should you guess for yp?
Answer: yp = At² + Bt + C (general second-degree polynomial).
5
For y'' - 4y = e2t, what form should you guess for yp (note: r=2 is a root)?
Answer: Since e2t is already in yh (root r=2 is simple), multiply by t: yp = Ate2t.
6
Solve y'' - y = 2et.
Answer: yh = c1et+c2e-t. r=1 is a root, so guess yp=Atet. yp'=A(1+t)et, yp''=A(2+t)et. A(2+t)et-Atet=2et. 2A=2, A=1. yp=tet.
7
What is the Wronskian of et and tet?
Answer: W = et(et+tet) - tet(et) = e2t + te2t - te2t = e2t.
8
True or false: Variation of parameters can only be used with constant-coefficient equations.
Answer: False. Variation of parameters works for any second-order linear ODE y''+p(t)y'+q(t)y=g(t), as long as you know y1 and y2.
9
What determines whether the solution to a homogeneous equation oscillates?
Answer: Complex roots in the characteristic equation. When D < 0, roots are α ± βi, and the solution contains sin and cos terms.
10
Solve y'' + y' = 0, y(0) = 4, y'(0) = -2.
Answer: r(r+1)=0, r=0,-1. y=c1+c2e-t. y(0)=c1+c2=4. y'=-c2e-t, y'(0)=-c2=-2, c2=2, c1=2. y=2+2e-t.