Section 5.2 Problem 17: Solve the initial value problem. 4y'' - 12y' + 9y = 0 y(0) = 3 y'(0) = (5)/(2) ​

Question

Section 5.2 Problem 17:

Solve the initial value problem.

`4y'' - 12y' + 9y = 0`

`y(0) = 3`

`y'(0) = (5)/(2)`
​

Answer 1

This DE has characteristic equation

`4r^2 - 12r + 9r = (2r - 3)^2 = 0`

with a repeated root at r = 3/2. Then the characteristic solution is

`y_c = C_1 e^(\frac32 x) + C_2 x e^(\frac32 x)`

which has derivative

`{y_c}' = \frac{3C_1}2 e^(\frac32 x) + \frac{3C_2}2 x e^(\frac32x) + C_2 e^(\frac32 x)`

Use the given initial conditions to solve for the constants:

`y(0) = 3 \implies 3 = C_1`

`y'(0) = \frac52 \implies \frac52 = \frac{3C_1}2 + C_2 \implies C_2 = -2`

and so the particular solution to the IVP is

`\boxed{y(x) = 3 e^(\frac32 x) - 2 x e^(\frac32 x)}`