Question

Verify that the indicated family of functions is a solution of the given differential equation. Assume an appropriate interval I of definition for each solution.

d^2y/ dx^2 − 6 dy/dx + 9y = 0;    y = c₁e^3x + c₂xe^3x  When y = c₁e^3x + c₂xe^3x,

dy dx	=
d2y dx2	=	.

Thus, in terms of x,

d^2y/dx^2− 6 dy/dx + 9y	=	+ 9(c1e3x + c2xe3x)
	=	.

2) In this problem, y = c₁e^x + c₂e^−x is a two-parameter family of solutions of the second-order DE

y'' − y = 0. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions.

y(−1) = 7,    y'(−1) = −7

y =

Answer 1