In class we talked about the reduction of orders. Namely, if we know y₁ for a second order equation, we can find y2 by assuming y₁ = v(t)y₁ and solving a first order ODE. Given the following equation and y₁: t²y" + 3ty' + y = 0, t>0; y₁(t) = t−1 (a) Find y2 using the reduction of orders. (b) Recall the Abel's Theorem: If y₁ and y2 are solutions of the second-order linear differential equation y" + p(t)y' +q(t)y = 0, - where p and q are continuous, then the Wronskian W(y1, y2, t) = Y1Y2 — Y2y'₁ is given by W(y1,y2,t) = c exp (- Sp(t)dt) where c is a certain constant that depends on y₁ and y2, but not on t. Further, W(y1, y2, t) either is zero for all t (if c = 0) or else is never zero (if c 0). This theorem gives you another way of finding y2. Can you find it using this Theorem?
In class we talked about the reduction of orders. Namely, if we know y₁ for a second order equation, we can find y2 by assuming y₁ = v(t)y₁ and solving a first order ODE. Given the following equation and y₁: t²y" + 3ty' + y = 0, t>0; y₁(t) = t−1 (a) Find y2 using the reduction of orders. (b) Recall the Abel's Theorem: If y₁ and y2 are solutions of the second-order linear differential equation y" + p(t)y' +q(t)y = 0, - where p and q are continuous, then the Wronskian W(y1, y2, t) = Y1Y2 — Y2y'₁ is given by W(y1,y2,t) = c exp (- Sp(t)dt) where c is a certain constant that depends on y₁ and y2, but not on t. Further, W(y1, y2, t) either is zero for all t (if c = 0) or else is never zero (if c 0). This theorem gives you another way of finding y2. Can you find it using this Theorem?
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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Transcribed Image Text:In class we talked about the reduction of orders. Namely, if we know
y₁ for a second order equation, we can find y2 by assuming y₁ = v(t)y₁ and solving a first
order ODE. Given the following equation and y₁:
t²y" + 3ty' + y = 0, t>0; y₁(t) = t−1
(a) Find y2 using the reduction of orders.
(b) Recall the Abel's Theorem:
If y₁ and y2 are solutions of the second-order linear differential equation
y" + p(t)y' +q(t)y = 0,
-
where p and q are continuous, then the Wronskian W(y1, y2, t) = Y1Y2 — Y2y'₁ is given by
W(y1,y2,t) = c exp
(- Sp(t)dt)
where c is a certain constant that depends on y₁ and y2, but not on t. Further, W(y1, y2, t)
either is zero for all t (if c = 0) or else is never zero (if c 0).
This theorem gives you another way of finding y2. Can you find it using this Theorem?
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