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2. Suppose that y₁ is a solution for of the homogeneous SOLDE " + p(t)y' + q(t)y = 0. (a) Explain how to find a non constant function tv(t) such that y2(t) = v(t)y₁(t) is also a solution. (b) Explain why y₁ and y₂ are linearly independent. Suppose that y, is a solution for of the homogeneous SOLDE "I y" +p(t)y +q(t)y = 0. (a) Explain how to find a non constant function t| → v(t) -> such that y2 (t) = v(t)y, (t) is also a solution. (b) Explain why y, and y2 linearly independent. The partial are solution is given below, please give all steps clearly with all math: For (a) you have to derive the method of reduction of order. Indeed, if we want y2 = v(t)y 1(t) to be a solution we get a homogeneous FOLDE with a solution given by v(t) = (-Sp)/(y1(t))^2 dt. For (b), one way is to explain why the v obtained in (a) is not constant and hence y2 is not a scalar multiple of y1. Another possible solution for (b) is to compute the Wronskian of y1 and y2 vy1 and explain why is never 0 =

2. Suppose that y₁ is a solution for of the homogeneous SOLDE " + p(t)y' + q(t)y = 0. (a) Explain how to find a non constant function tv(t) such that y2(t) = v(t)y₁(t) is also a solution. (b) Explain why y₁ and y₂ are linearly independent. Suppose that y, is a solution for of the homogeneous SOLDE "I y" +p(t)y +q(t)y = 0. (a) Explain how to find a non constant function t| → v(t) -> such that y2 (t) = v(t)y, (t) is also a solution. (b) Explain why y, and y2 linearly independent. The partial are solution is given below, please give all steps clearly with all math: For (a) you have to derive the method of reduction of order. Indeed, if we want y2 = v(t)y 1(t) to be a solution we get a homogeneous FOLDE with a solution given by v(t) = (-Sp)/(y1(t))^2 dt. For (b), one way is to explain why the v obtained in (a) is not constant and hence y2 is not a scalar multiple of y1. Another possible solution for (b) is to compute the Wronskian of y1 and y2 vy1 and explain why is never 0 =

Advanced Engineering Mathematics
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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2. Suppose that y₁ is a solution for of the homogeneous SOLDE " + p(t)y' + q(t)y = 0. (a) Explain how to find a non constant function tv(t) such that y2(t) = v(t)y₁(t) is also a solution. (b) Explain why y₁ and y₂ are linearly independent.
Transcribed Image Text:2. Suppose that y₁ is a solution for of the homogeneous SOLDE " + p(t)y' + q(t)y = 0. (a) Explain how to find a non constant function tv(t) such that y2(t) = v(t)y₁(t) is also a solution. (b) Explain why y₁ and y₂ are linearly independent.
Suppose that y, is a solution for of the homogeneous SOLDE "I y" +p(t)y +q(t)y = 0. (a) Explain how to find a non constant function t| → v(t) -> such that y2 (t) = v(t)y, (t) is also a solution. (b) Explain why y, and y2 linearly independent. The partial are solution is given below, please give all steps clearly with all math: For (a) you have to derive the method of reduction of order. Indeed, if we want y2 = v(t)y 1(t) to be a solution we get a homogeneous FOLDE with a solution given by v(t) = (-Sp)/(y1(t))^2 dt. For (b), one way is to explain why the v obtained in (a) is not constant and hence y2 is not a scalar multiple of y1. Another possible solution for (b) is to compute the Wronskian of y1 and y2 vy1 and explain why is never 0 =
Transcribed Image Text:Suppose that y, is a solution for of the homogeneous SOLDE "I y" +p(t)y +q(t)y = 0. (a) Explain how to find a non constant function t| → v(t) -> such that y2 (t) = v(t)y, (t) is also a solution. (b) Explain why y, and y2 linearly independent. The partial are solution is given below, please give all steps clearly with all math: For (a) you have to derive the method of reduction of order. Indeed, if we want y2 = v(t)y 1(t) to be a solution we get a homogeneous FOLDE with a solution given by v(t) = (-Sp)/(y1(t))^2 dt. For (b), one way is to explain why the v obtained in (a) is not constant and hence y2 is not a scalar multiple of y1. Another possible solution for (b) is to compute the Wronskian of y1 and y2 vy1 and explain why is never 0 =
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