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5. Consider the functions y₁ = t and y₂ = tlnt. (a) Find the Wronskian determinant W (y₁, y2) for all t > 0. (b) Verify that y₁ and y2 are solutions of t²y" - ty' + y = 0 for t > 0. Moreover, show that 91, 92 form a fundamental set of solutions for the ODE. (c) Solve t2y"-ty' + y = 0, y(3) = 2, y'(3) = -1.

5. Consider the functions y₁ = t and y₂ = tlnt. (a) Find the Wronskian determinant W (y₁, y2) for all t > 0. (b) Verify that y₁ and y2 are solutions of t²y" - ty' + y = 0 for t > 0. Moreover, show that 91, 92 form a fundamental set of solutions for the ODE. (c) Solve t2y"-ty' + y = 0, y(3) = 2, y'(3) = -1.

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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Please answer the following differential equations question.
### Problem 5: Consider the functions \( y_1 = t \) and \( y_2 = t \ln t \). #### (a) Find the Wronskian determinant \( W(y_1, y_2) \) for all \( t > 0 \). #### (b) Verify that \( y_1 \) and \( y_2 \) are solutions of the differential equation \( t^2 y'' - t y' + y = 0 \) for \( t > 0 \). Moreover, show that \( y_1, y_2 \) form a fundamental set of solutions for the Ordinary Differential Equation (ODE). #### (c) Solve the ODE \( t^2 y'' - t y' + y = 0 \) with the initial conditions \( y(3) = 2 \), \( y'(3) = -1 \).
Transcribed Image Text:### Problem 5: Consider the functions \( y_1 = t \) and \( y_2 = t \ln t \). #### (a) Find the Wronskian determinant \( W(y_1, y_2) \) for all \( t > 0 \). #### (b) Verify that \( y_1 \) and \( y_2 \) are solutions of the differential equation \( t^2 y'' - t y' + y = 0 \) for \( t > 0 \). Moreover, show that \( y_1, y_2 \) form a fundamental set of solutions for the Ordinary Differential Equation (ODE). #### (c) Solve the ODE \( t^2 y'' - t y' + y = 0 \) with the initial conditions \( y(3) = 2 \), \( y'(3) = -1 \).
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