Introduction 15In exercises 18-26 v erify that the given function or functions is a
solution of the give n diffe r ential equation and s p ecify the interval
or intervals on which the solution exists.18. y" - 3y' + 2y = O; Y2(x) = e 2 x
- x^2 y" - 2y = O; Yi(x ) = x^2 - l/x
- y' + 1/(2y) = O; Yi(x ) = J3=X
2 1. y' - y / x = 1; Yi ( x ) = x ln x
y' - 2J[Yf = O; Yi (x ) = xlxl
x^2 dy + 2xydx = O; Yi (x ) = -1/x^2
24. y' - y^2 = 1; Yi (x ) = tanx
- 2x^2 y" + x y' - y = O; Yi (x ) = x, Y2(x ) = 1/ y'x
- x y' - sinx = O;
r sintdt
Yi (x ) =Jo - t- , ( )
_ 1 71" sin t dt
Y2 X -- --
x tIn exercises 27-30 find the value or values of r for which the func-
tion y (x ) = erx is a solution of the give n diffe r entia l equation.
y' + 3y = 0 28. y" - 3y' - lOy = 0
y" + 2y' + y = 0 3 0. y"' - 7y" + 12y' = 0
In exercises 31-34 find the value or values of r for which the func-
tion y (x)=x r is a solution of the give n differentia l equation.
3 1. 2xy' - y = 0 32. x^2 y" - xy' = 0
- x^2 y" + 6x y' + 4y = 0 34. x^2 y" - 5x y' + 9y =^0