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6. 5. Find the general solution of the equation xy" + 2y' + xy = 0, if we know that y₁ = solution. COS X X is a

6. 5. Find the general solution of the equation xy" + 2y' + xy = 0, if we know that y₁ = solution. COS X X is a

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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4/5: Please solve the following differential equations marked with a red circle.
**Question 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) \( y_1 = 2e^{3t} - 5e^{-t}, \quad y_2 = 11e^{-t} \) (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) \( y_1 = e^{3t}, \quad y_2 = te^{3t} \) **Question 5:** Find the general solution of the equation \( xy'' + 2y' + xy = 0 \), if we know that \( y_1 = \frac{\cos x}{x} \) is a solution.
Transcribed Image Text:**Question 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) \( y_1 = 2e^{3t} - 5e^{-t}, \quad y_2 = 11e^{-t} \) (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) \( y_1 = e^{3t}, \quad y_2 = te^{3t} \) **Question 5:** Find the general solution of the equation \( xy'' + 2y' + xy = 0 \), if we know that \( y_1 = \frac{\cos x}{x} \) is a solution.
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