Page 291 - Schaum's Outline of Differential Equations

274 LINEAR DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS [CHAP. 27

Substituting these results into Eq. (_/), we obtain the solution as

27.25. Show that the method of undetermined coefficients cannot be used to obtain a particular solution of
y" + xy = 2.
By the method of undetermined coefficients, we assume a particular solution of the form y p = A^, where m
might be zero if the simple guess y p = A Q does not require modification (see Chapter 11). Substituting y p into the
differential equation, we find

Regardless of the value of m, it is impossible to assign A Q any constant value that will satisfy (_/). It follows that the
method of undetermined coefficients is not applicable.
One limitation on the method of undetermined coefficients is that it is only valid for linear equations with
constant coefficients.

Supplementary Problems

In Problems 27.26 through 27.34, determine whether the given values of x are ordinary points or singular points of the given
differential equations.

2
2
27.26. x= l;y" + 3y' + 2xy = 0 27.27. x = 2;(x- 2)y" + 3(x -3x + 2)/ +(x- 2) y = 0
27.28. x = Q;(x + l)y"+-y' + xy = Q 27.29. x = -l;(x + l)y' + -y'+xy = 0
X x
i
27.30. x = 0; x^y" + y = 0 27.31. x = 0;x y" + xy = 0
2
x
27.32. x = 0; e y" + (sin x)y' + xy = 0 27.33. x = -l;(x+ ify" + (x -l)(x + I)/ +(x-l)y = 0
2
27.34. x = 2; x\£ - 4)y" +(x+l)y'+ (x -3x + 2)y = 0
27.35. Find the general solution near x = 0 of y" - y' = 0. Check your answer by solving the equation by the method of
Chapter 9 and then expanding the result in a power series about x = 0.

In Problems 27.36 through 27.47, find (a) the recurrence formula and (b) the general solution of the given differential
equation by the power series method around, the given value of x.

27.36. x = 0; y" + xy = 0 27.37. x = 0; y"-2xy'-2y = 0

2
2
27.38. x = 0; /' + x y' + 2xy = 0 27.39. x = 0; y" - x y' - y = 0
2
2
27.40. x = 0; y" + 2x y = 0 27.41. x = 0; (x -l)y" + xy'-y = 0
27.42. x = 0; y"-xy = 0 27.43. x=l; y"-xy = 0
2
2
27.44. x = -2; y" -x y ' + (x + 2)y = 0 27.45. x = 0; (x +4)y" + y = x
2
27.46. x=l; y"-(x- I)/ = x - 2x 27.47. x = Q- y"-xy' = e- x
2
27.48. Use the Taylor series method described in Problem 27.23 to solve /' - 2xy' + x y = 0; y(0) = 1, /(O) = -1.
2
27.49. Use the Taylor series method described in Problem 27.23 to solve /' - 2xy = x ; (l) = 0, /(I) = 2.
y