Consider the variable coefficient linear second order non-homogeneous ODE x?y" – 2xy + (² + 2)y = 3x³ for r> 0. (1) %3| 1. Write down the associated homogeneous equation. 2. Show that yı(x) = x sin x and y2(x) = x cos a are solutions of the associated homogeneous equation. 3. Use the Wronskian test to determine if y1 and y2 are linearly independent for a > 0. 4. Write down the solution Yn(x) of the associated homogeneous equation. 5. Use the formulas in the course notes for the method of variation of parameters to find a particular solution yp(x) of equation (1). Hint: you will need to re-write the ODE in standard form in order to identify the non-homogeneous term in the formulas. 6. Write down the general solution to equation (1). 7. Find the solution to equation (1) that satisfies the initial conditions y(T/2) and y'(7/2) = 0. 2
Consider the variable coefficient linear second order non-homogeneous ODE x?y" – 2xy + (² + 2)y = 3x³ for r> 0. (1) %3| 1. Write down the associated homogeneous equation. 2. Show that yı(x) = x sin x and y2(x) = x cos a are solutions of the associated homogeneous equation. 3. Use the Wronskian test to determine if y1 and y2 are linearly independent for a > 0. 4. Write down the solution Yn(x) of the associated homogeneous equation. 5. Use the formulas in the course notes for the method of variation of parameters to find a particular solution yp(x) of equation (1). Hint: you will need to re-write the ODE in standard form in order to identify the non-homogeneous term in the formulas. 6. Write down the general solution to equation (1). 7. Find the solution to equation (1) that satisfies the initial conditions y(T/2) and y'(7/2) = 0. 2
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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Transcribed Image Text:Consider the variable coefficient linear second order non-homogeneous ODE
x?y" – 2xy + (² + 2)y = 3x³ for r> 0.
(1)
1. Write down the associated homogeneous equation.
2. Show that yı (x) = x sin x and y2(x) = x cos x are solutions of the associated homogeneous
equation.
3. Use the Wronskian test to determine if y1 and y2 are linearly independent for a > 0.
4. Write down the solution Yn(x) of the associated homogeneous equation.
5. Use the formulas in the course notes for the method of variation of parameters to find
a particular solution yp(x) of equation (1). Hint: you will need to re-write the ODE in
standard form in order to identify the non-homogeneous term in the formulas.
6. Write down the general solution to equation (1).
7. Find the solution to equation (1) that satisfies the initial conditions
y(T/2)
and y'(7/2) = 0.
2
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