Page 98 - Schaum's Outline of Differential Equations

CHAP. 8] LINEAR DIFFERENTIAL EQUATIONS: THEORY OF SOLUTIONS 81

To prove that it is the general solution, we must show that every solution of \-(y) = 0 (x) is of the form (8.9).
Let y be any solution of \-(y) = 0 (x) and set z = y — y p. Then

so that z is a solution to the homogeneous equation \-(y) = 0. Since i = y — y p, it follows that y = i + y p, where z is
a solution of \-(y) = 0.

Supplementary Problems

8.33. Determine which of the following differential equations are linear:

(a) /' + xy' + 2y = 0 (b) y'" -y = x

2
2
x
(c) / + 5y = 0 (d) y (4) + x y'" + xy" - e y' + 2y = x + x+l
(e) /' + 2xy' + y = 4xy 2 (/) y'-2y = xy
2
(g) y" + yy' = x 2 (K) y'" + (x -l)y"-2y' + y = 5smx
(i) y' + y(sin x)=x (j) y' + x(sin y)=x

x
(k) y"+ ey = 0 (I) y" + e = 0
8.34. Determine which of the linear differential equations in Problem 8.33 are homogeneous.

8.35. Determine which of the linear differential equations in Problem 8.33 have constant coefficients.

In Problems 8.36 through 8.49, find the Wronskians of the given sets of functions and, where appropriate, use that information
to determine whether the given sets are linearly independent.
2
8.36. {3x, 4x} 8.37. [x , x}
2
3
8.38. {x , x } 8.39. {x , x}
3
8.40. [x ,5} 8.41. {x , -x }
2
2
2
3
2
2x
8.42. {e , e- *} 8.43. {e , e *}
2x
8.44. {Se *, Se *} 8.45. {jc, 1, 2x-7}
2
2
2
8.46. {x+l,x 2 + x,2x -x-3} 8.47. {je , x , x }
2
3
4
2
8.48. {e-\ <•?, *} 8.49. {sin jc, 2 cos jc, 3 sin x + cos jc}
e
8.50. Prove directly that the set given in Problem 8.36 is linearly dependent.
8.51. Prove directly that the set given in Problem 8.41 is linearly dependent.
8.52. Prove directly that the set given in Problem 8.44 is linearly dependent.
8.53. Prove directly that the set given in Problem 8.45 is linearly dependent.
8.54. Prove directly that the set given in Problem 8.46 is linearly dependent.