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3. Consider the non-homogeneous Cauchy-Euler equation ax²y" + bxy' + cy = f(x) (a) Let y₁(x) and 32(2) be solutions to the associated homogeneous equation, and let y,(r) be a solution to the non-homogeneous equation. Is y(x) = C₁y₁(x) + C2y2(x) + yp(x) the general solution to the non-homogeneous ODE? Prove or provide a counterexample. (b) Solve x²y" + xy' - 9y = 5e²¹ using any method.

3. Consider the non-homogeneous Cauchy-Euler equation ax²y" + bxy' + cy = f(x) (a) Let y₁(x) and 32(2) be solutions to the associated homogeneous equation, and let y,(r) be a solution to the non-homogeneous equation. Is y(x) = C₁y₁(x) + C2y2(x) + yp(x) the general solution to the non-homogeneous ODE? Prove or provide a counterexample. (b) Solve x²y" + xy' - 9y = 5e²¹ using any method.

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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3. Consider the non-homogeneous Cauchy-Euler equation ax²y" + bry' + cy = f(x) (a) Let y₁(x) and y2 (r) be solutions to the associated homogeneous equation, and let y(x) be a solution to the non-homogeneous equation. Is y(x) = C₁y₁(x) + C2Y2(x) +Yp(x) the general solution to the non-homogeneous ODE? Prove or provide a counterexample. (b) Solve x²y" + xy' - 9y = 5e²¹ using any method.
Transcribed Image Text:3. Consider the non-homogeneous Cauchy-Euler equation ax²y" + bry' + cy = f(x) (a) Let y₁(x) and y2 (r) be solutions to the associated homogeneous equation, and let y(x) be a solution to the non-homogeneous equation. Is y(x) = C₁y₁(x) + C2Y2(x) +Yp(x) the general solution to the non-homogeneous ODE? Prove or provide a counterexample. (b) Solve x²y" + xy' - 9y = 5e²¹ using any method.
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