4.9E: Reduction of Order (Exercises)
- Page ID
- 43303
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Q4.9.1
In Exercises 4.9.1-4.9.17 find the general solution, given that \(y_1\) satisfies the complementary equation. As a byproduct, find a fundamental set of solutions of the complementary equation.
1. \((2x+1)y''-2y'-(2x+3)y=(2x+1)^2; \quad y_1=e^{-x}\)
2. \(x^2y''+xy'-y={4\over x^2}; \quad y_1=x\)
3. \(x^2y''-xy'+y=x; \quad y_1=x\)
4. \(y''-3y'+2y={1\over1+e^{-x}}; \quad y_1=e^{2x}\)
5. \(y''-2y'+y=7x^{3/2}e^x; \quad y_1=e^x\)
6. \(4x^2y''+(4x-8x^2)y'+(4x^2-4x-1)y=4x^{1/2}e^x(1+4x); \quad y_1=x^{1/2}e^x\)
7. \(y''-2y'+2y=e^x\sec x; \quad y_1=e^x\cos x\)
8. \(y''+4xy'+(4x^2+2)y=8e^{-x(x+2)}; \quad y_1=e^{-x^2}\)
9. \(x^2y''+xy'-4y=-6x-4; \quad y_1=x^2\)
10. \(x^2y''+2x(x-1)y'+(x^2-2x+2)y=x^3e^{2x}; \quad y_1=xe^{-x}\)
11. \(x^2y''-x(2x-1)y'+(x^2-x-1)y=x^2e^x; \quad y_1=xe^x\)
12. \((1-2x)y''+2y'+(2x-3)y=(1-4x+4x^2)e^x; \quad y_1=e^x\)
13. \(x^2y''-3xy'+4y=4x^4; \quad y_1=x^2\)
14. \(2xy''+(4x+1)y'+(2x+1)y=3x^{1/2}e^{-x}; \quad y_1=e^{-x}\)
15. \(xy''-(2x+1)y'+(x+1)y=-e^x; \quad y_1=e^x\)
16. \(4x^2y''-4x(x+1)y'+(2x+3)y=4x^{5/2}e^{2x}; \quad y_1=x^{1/2}\)
17. \(x^2y''-5xy'+8y=4x^2; \quad y_1=x^2\)
Q4.9.2
In Exercises 4.9.18-4.9.30 find a fundamental set of solutions, given that \(y_{1}\) is a solution.
18. \(xy''+(2-2x)y'+(x-2)y=0; \quad y_1=e^x\)
19. \(x^2y''-4xy'+6y=0; \quad y_1=x^2\)
20. \(x^2(\ln |x|)^2y''-(2x \ln |x|)y'+(2+\ln |x|)y=0; \quad y_1=\ln |x|\)
21. \(4xy''+2y'+y=0; \quad y_1=\sin \sqrt{x}\)
22. \(xy''-(2x+2)y'+(x+2)y=0; \quad y_1=e^x\)
23. \(x^2y''-(2a-1)xy'+a^2y=0; \quad y_1=x^a\)
24. \(x^2y''-2xy'+(x^2+2)y=0; \quad y_1=x \sin x\)
25. \(xy''-(4x+1)y'+(4x+2)y=0; \quad y_1=e^{2x}\)
26. \(4x^2(\sin x)y''-4x(x\cos x+\sin x)y'+(2x\cos x+3\sin x)y=0; \quad y_1=x^{1/2}\)
27. \(4x^2y''-4xy'+(3-16x^2)y=0; \quad y_1=x^{1/2}e^{2x}\)
28. \((2x+1)xy''-2(2x^2-1)y'-4(x+1)y=0; \quad y_1=1/x\)
29. \((x^2-2x)y''+(2-x^2)y'+(2x-2)y=0; \quad y_1=e^x\)
30. \(xy''-(4x+1)y'+(4x+2)y=0; \quad y_1=e^{2x}\)
Q4.9.3
In Exercises 4.9.31-4.9.33 solve the initial value problem, given that \(y_{1}\) satisfies the complementary equation.
31. \(x^2y''-3xy'+4y=4x^4,\quad y(-1)=7,\quad y'(-1)=-8; \quad y_1=x^2\)
32. \((3x-1)y''-(3x+2)y'-(6x-8)y=0, \quad y(0)=2,\; y'(0)=3; \quad y_1=e^{2x}\)
33. \((x+1)^2y''-2(x+1)y'-(x^2+2x-1)y=(x+1)^3e^x, \quad y(0)=1,\quad y'(0)=~-1; \quad y_1=(x+1)e^x\)
Q4.9.4
In Exercises 4.9.34 and 4.9.35 solve the initial value problem and graph the solution, given that \(y_{1}\) satisfies the complementary equation.
34. \(x^2y''+2xy'-2y=x^2, \quad y(1)={5\over4},\; y'(1)={3\over2}; \quad y_1=x\)
35. \((x^2-4)y''+4xy'+2y=x+2, \quad y(0)=-{1\over3},\quad y'(0)=-1; \quad y_1={1\over x-2}\)