(b) Show that x₁(t) = eat and x₂(t) = t + ß are solutions of the homogeneous equation tx" (t) − (1 +t)x'(t) + x(t) = 0, te (0, ∞), for suitable constants a and B which should be determined. Apply the method of part (a) (variation of parameters) to find the general solution to the second-order linear ODE tx"(t) − (1+t)x' (t) + x(t) = t²e²t, te (0, ∞). Hint: You may find the integral ft ses ds = (t − 1)et + c useful.
(b) Show that x₁(t) = eat and x₂(t) = t + ß are solutions of the homogeneous equation tx" (t) − (1 +t)x'(t) + x(t) = 0, te (0, ∞), for suitable constants a and B which should be determined. Apply the method of part (a) (variation of parameters) to find the general solution to the second-order linear ODE tx"(t) − (1+t)x' (t) + x(t) = t²e²t, te (0, ∞). Hint: You may find the integral ft ses ds = (t − 1)et + c useful.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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PLEASE ANSWER QUESTION CLEARLY
Transcribed Image Text:(b) Show that x₁(t)
=
eat and x₂(t) = t + ß are solutions of the homogeneous equation
tx" (t) — (1+t)x' (t) + x(t) = 0,
te (0, ∞),
for suitable constants a and 3 which should be determined.
Apply the method of part (a) (variation of parameters) to find the general solution to
the second-order linear ODE
tx"(t)
(1+t)x'(t) + x(t) = t²e²t,
t = (0, ∞0).
Hint: You may find the integral ft se³ ds = (t − 1)et + c useful.
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