Sample Solutions to Quiz 3 for MATH3270A − y

Comments

Transcription

Sample Solutions to Quiz 3 for MATH3270A − y
Sample Solutions to Quiz 3 for MATH3270A
October 31,2013
1.Find the general solution of the ODE:
y (3) − y (2) + y 0 − Y = e−t sin t.
Answer: The characteristic equation is:
r3 − r2 + r − 1 = 0
The roots are r1 = 1,r2 = i,r3 = −i.The general solution of the homogeneous equation is:
yh = c1 et + c2 cos t + c3 sin t
A particular solution has the form yp = Ae−t sin t + Be−t cos t. Substitute it into the equation we get A = − 51 , B = 0. Thus the general
solution is
1
y = c1 et + c2 cos t + c3 sin t − e−t cos t
5
2.Show that W [1, sin2 t, cos 2t] ≡ 0 for all t.Can you prove this without
direct evaluation?
Answer: Since cos 2t = cos2 t − sin2 t = 1 − 2 sin2 t,thus 1,sin2 t,cos2 t
are linearly dependent,then Wronskian must be zero.
3.Find the Wronskian of a fundamental set of solutions to the ODE:
ty (3) + 2y (2) − 5y = 0, t > 0
Answer:
1
2
By Abel’s formula:
Z
W (t) = c exp[−
2
dt] = c exp[−2 ln |t|] = ct−2
t