Differential Equation: Need Help

Question

Determine the specific solution of the following initial-value problems. y^''-6y^'+9y=0, y(-2)=1 and y^'(-2)=0

Answer 1

y^''-6y^'+9y=0, y(-2)=1 and y^'(-2)=0

k²-6k+9=0 (this characteristics equation)

k=3 is a repeated root (twice). So, the general solution is y(x)=(A+Bx)e^3x

y'=(B+3A+3Bx)e^3x

y(-2)=1, y'(-2)=0

A-2B=e⁶

3A-5B=0

A=-5e⁶, B=-3e⁶

y=-(5+3x)e^3x+6