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8. xy" - xy + y = 0 y₁(x) = x

8. xy" - xy + y = 0 y₁(x) = x

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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can you please explain how to get the result for number 8 please. I am so lost

**PROBLEMS: Section 3.3** For Problems 1–10, a differential equation and one solution are given. Use d'Alembert's reduction of order method to find a second linearly independent solution. What is the general solution of the differential equation? **Differential Equation - Solution** 1. \( y'' - y = 0 \) \( y_1(x) = e^x \) 2. \( y'' + y = 0 \) \( y_1(x) = \sin x \) 3. \( y'' - 4y' + 4y = 0 \) \( y_1(x) = e^{2x} \) 4. \( y'' + y' = 0 \) \( y_1(x) = 1 \) 5. \( y'' + y' = 0 \) \( y_1(x) = 1 \) 6. \( xy'' - 2(x + 1)y' + 4y = 0 \) \( y_1(x) = e^x \) 7. \( x^2y'' - 6y = 0 \) \( y_1(x) = x^3 \) 8. \( y'' - xy' + y = 0 \) \( y_1(x) = x \) 9. \( (x^2 + 1)y'' - 2xy' + 2y = 0 \) \( y_1(x) = x \)
Transcribed Image Text:**PROBLEMS: Section 3.3** For Problems 1–10, a differential equation and one solution are given. Use d'Alembert's reduction of order method to find a second linearly independent solution. What is the general solution of the differential equation? **Differential Equation - Solution** 1. \( y'' - y = 0 \) \( y_1(x) = e^x \) 2. \( y'' + y = 0 \) \( y_1(x) = \sin x \) 3. \( y'' - 4y' + 4y = 0 \) \( y_1(x) = e^{2x} \) 4. \( y'' + y' = 0 \) \( y_1(x) = 1 \) 5. \( y'' + y' = 0 \) \( y_1(x) = 1 \) 6. \( xy'' - 2(x + 1)y' + 4y = 0 \) \( y_1(x) = e^x \) 7. \( x^2y'' - 6y = 0 \) \( y_1(x) = x^3 \) 8. \( y'' - xy' + y = 0 \) \( y_1(x) = x \) 9. \( (x^2 + 1)y'' - 2xy' + 2y = 0 \) \( y_1(x) = x \)
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