SECTION 5.5 Differential Equations 307
- Show thaty =Ke^2 x−^14 (2x+ 1) is a solution of the linear ODE
y′= 2y+xfor any value of the constantK. - Find a first-order linear ODE having y = x^2 + 1 as a solution.
(There are many answers.) - In each case below, verify that the linear ODE has the given func-
tion as a solution.
(a) xy′+y= 3x^2 , y=x^2.
(b) y′+ 2xy= 0, y=e−x
2
.
(c) 2x^2 y′′+ 3xy′−y= 0, y=
√
x, x >0.- Consider then-th order linear ODE withconstant coefficients:
y(n)+an− 1 y(n−1)+···+a 1 y′+y = 0. (5.1)Assume that the associatedcharacteristic polynomial