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+ 4. In each case, find a second order, linear, homogeneous ODE that has the given pair of functions as its solutions. Then show that the pair forms a fundamental set of solutions. (a) y₁ = 2e³t - 5e-t, y₂ = 11e-t (b) y₁ = cos(πt), y₂ = sin(nt) Y2 (c) y₁ = e²t cos(5t), y2 = e²t sin(5t) (d) y₁ = e³t, y₂ = te³t

+ 4. In each case, find a second order, linear, homogeneous ODE that has the given pair of functions as its solutions. Then show that the pair forms a fundamental set of solutions. (a) y₁ = 2e³t - 5e-t, y₂ = 11e-t (b) y₁ = cos(πt), y₂ = sin(nt) Y2 (c) y₁ = e²t cos(5t), y2 = e²t sin(5t) (d) y₁ = e³t, y₂ = te³t

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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4. Please solve the following differential equation.

**Problem 4:** For each case, find a second-order, linear, homogeneous ordinary differential equation (ODE) with the given pair of functions as its solutions. Then demonstrate that this pair forms a fundamental set of solutions. **(a)** \( y_1 = 2e^{3t} - 5e^{-t}, \quad y_2 = 11e^{-t} \) **(b)** \( y_1 = \cos(\pi t), \quad y_2 = \sin(\pi t) \) **(c)** \( y_1 = e^{2t} \cos(5t), \quad y_2 = e^{2t} \sin(5t) \) **(d)** \( y_1 = e^{3t}, \quad y_2 = te^{3t} \)
Transcribed Image Text:**Problem 4:** For each case, find a second-order, linear, homogeneous ordinary differential equation (ODE) with the given pair of functions as its solutions. Then demonstrate that this pair forms a fundamental set of solutions. **(a)** \( y_1 = 2e^{3t} - 5e^{-t}, \quad y_2 = 11e^{-t} \) **(b)** \( y_1 = \cos(\pi t), \quad y_2 = \sin(\pi t) \) **(c)** \( y_1 = e^{2t} \cos(5t), \quad y_2 = e^{2t} \sin(5t) \) **(d)** \( y_1 = e^{3t}, \quad y_2 = te^{3t} \)
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