Given the second order homogeneous constant coefficient equation y′′−4y′−12y=0 1) the characteristic polynomial ar2+br+c is r^2-4r-12 . 2) The roots of auxiliary equation are (enter answers as a comma separated list). 3) A fundamental set of solutions is (enter answers as a comma separated list). 4) Given the initial conditions y(0)=1 and y′(0)=−26 find the unique solution to the IVP y= .

The initial value problem

Step-by-step explanation:

Step1:-

a) Given second order homogenous constant co-efficient equation

Given equation in the operator form is

Step 2:-

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

Step 3:-

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

answer are

step-by-step explanation:

im here

step-by-step explanation: