5.7.1: Cauchy-Euler Equations (Exercises)

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In Exercises 1-40 find the general solution of the given equation on \((0,\infty)\).

1. \(x^2y''+7xy'+8y=0\)

2. \(x^2y''-7xy'+7y=0\)

3. \(x^2y''-xy'+y=0\)

4. \(x^2y''+5xy'+4y=0\)

5. \(x^2y''+xy'+y=0\)

6. \(x^2y''-3xy'+13y=0\)

7. \(x^2y''+3xy'-3y=0\)

8. \(12x^2y''-5xy''+6y=0\)

9. \(4x^2y''+8xy'+y=0\)

10. \(3x^2y''-xy'+y=0\)

11. \(2x^2y''-3xy'+2y=0\)

12. \(x^2y''+3xy'+5y=0\)

13. \(9x^2y''+15xy'+y=0\)

14. \(x^2y''-xy'+10y=0\)

15. \(x^2y''-6y=0\)

16. \(2x^2y''+3xy'-y=0\)

17. \(x^2y''-3xy'+4y=0\)

18. \(2x^2y''+10xy'+9y=0\)

19. \(x^2y''-2y=0\)

20. \(xy''+y'=0\)

21. \(x^2y''+xy'+4y=0\)

22. \(x^{2} y^{\prime \prime}-3 x y^{\prime}-2 y=0\)

23. \(25x^2y''+25xy'+y=0\)

24. \(x^{2} y^{\prime \prime}+5 x y^{\prime}+4 y=0\)

25. \(3x^2y''+6xy'+y=0\)

26. \(x^{2} y^{\prime \prime}+3 x y^{\prime}-3 y=0\)

27. \(2 x^{2} y^{\prime \prime}+5 x y^{\prime}+y=0\)

28. \(4 x^{2} y^{\prime \prime}+y=0\)

29. \(xy''-4y'=x^4\)

30. \(x^2y''-xy'+y=2x\)

31. \(x^{2} y^{\prime \prime}- x y^{\prime}-3 y=2 x^{2}\)

32. \(x^{2} y^{\prime \prime}- x y^{\prime}-3 y=2 x^{3}\)

33. \(x^2y''-4xy'+6y=2x^4+x^2\)

34. \(x^2y''-3xy'+3y=2x^4e^x\)

35. \(x^2y''-xy'+y=\ln x\)

36. \(2 x^{2} y^{\prime \prime}+5 x y^{\prime}+y=x^{2}+x\)

37. \(x^{2} y^{\prime \prime}+5 x y^{\prime}+4 y=2 x^{3}\)

38. \(x^2y''+10xy'+8y=x^2\)

39. \(x^2y''-4xy'+6y=\ln x^2\)

40. \(x^2y''-3xy'+13y=4+3x\)

In Exercises 41-49 find the solution of the given initial value problem on \((0,\infty)\).

41. \(x^2y''+3xy'=0; \quad y(1)=0, \quad y'(1)=4\)

42. \(x^2y''-5xy'+8y=0; \quad y(2)=32, \quad y'(2)=0\)

43. \(x^2y''+xy'+y=0; \quad y(1)=1, \quad y'(1)=2\)

44. \(x^2y''-3xy'+4y=0; \quad y(1)=5, \quad y'(1)=3\)

45. \(x^2y''+3xy'+y=0; \quad y(1)=0, \quad y'(1)=1\)

46. \(xy''+y'=x; \quad y(1)=1, \quad y'(1)=-1/2\)

47. \(x^2y''-5xy'+8y=8x^6; \quad y({1\over 2})=0, \quad y'({1\over 2})=0\)

48. \(x^2y''-3xy'+3y=2x^4e^x; \quad y(1)=0, \quad y'(1)=0\)

49. \(x^2y''-xy'+y=\ln x; \quad y(1)=2, \quad y'(1)=2\)

50. Prove that if \(y=y(x)\) satisfies the Cauchy-Euler equation \[ax^2{d^2y\over dx^2}+bx{dy\over dx}+cy=0\tag{A}\] on \((0,\infty)\) then \(y=y(t)\), where \(t=-x\), satisfies A on \((-\infty,0)\).

In Exercises 51-52 find the solution of the given initial value problem on \((-\infty,0)\).

51. \(4x^2y''+y=0; \quad y(-1)=2, \quad y'(-1)=4\)

52. \(x^2y''-4xy'+6y=0; \quad y(-2)=8, \quad y'(-2)=0\)

53. A nontrivial solution of

\[P_0(x)y''+P_1(x)y'+P_2(x)y=0\nonumber\]

is said to be oscillatory on an interval \((a,b)\) if it has infinitely many zeros on \((a,b)\). Otherwise \(y\) is said to be nonoscillatory on \((a,b)\). Prove that the equation

\[x^2y''+ky=0 \quad (k=\; \mbox{constant})\nonumber\]

has oscillatory solutions on \((0,\infty)\) if and only if \(k>1/4\).