Find the general solution to y 12y 36y 0 Enter your answe

Find the general solution to y\" - 12y\' + 36y = 0. Enter your answer as y = ... .In your answer, use c_1 and c_2 to denote arbitrary constants and x the independent variable. Enter c_1 as c_1 and c_2 as c2. help (equations) Find the solution that satisfies the initial conditions y(0) = 7 and y\'(0) = 0. help (equations)

Solution

WE hav ea linear homogeneous equation with constant coefficients. So solution is of the form

y=exp(kx)

Subsituting gives

k^2-12k+36=0

k=6

So repeated roots

General solution is

y=c1 exp(6x)+c2 x exp(6x)

b)

y(0)=7=c1

y\'(0)=6c1+c2 =0

c2=-42

y=7 exp(6x)-42 x exp(6x)