Answer:
The initial value problem 
Step-by-step explanation:
<u>Step1:</u>-
a) Given second order homogenous constant co-efficient equation

Given equation in the operator form is 
<u>Step 2</u>:-
b) Let f(D) = 
Then the auxiliary equation is 
Find the factors of the auxiliary equation is

m(m-6) + 2(m-6) =0
m+2 =0 and m-6=0
m=-2 and m=6
The roots are real and different
The general solution 
the roots are 
The general solution of given differential equation is

<u>Step 3</u>:-
C) Given initial conditions are y(0) =1 and y1 (0) =-26
The general solution of given differential equation is
.....(1)
substitute x =0 and y(0) =1

.........(2)
Differentiating equation (1) with respective to 'x'

substitute x= o and y1 (0) =-26

.............(3)
solving (2) and (3) by using substitution method
in equation (3)

on simplification , we get


dividing by'8' we get 
substitute
in equation 
so 
now substitute
in general solution


now the initial value problem
