Question

The general solution of the nonhomogeneous equation y''-6y'-7=8e(-t)-7t-6 is

Answer 1

Correct question: Solve the following differential equation

$\quad D^{2}y-6Dy-7y=8e^{-x}-7x-6$

Solution:

The given differential equation is

$\quad D^{2}y-6Dy-7y=8e^{-x}-7x-6$

To find C. F.

The auxiliary equation is

$\quad m^{2}-6m-7=0$

$\Rightarrow (m+1)(m-7)=0$

$\Rightarrow m=-1,\:7$

C. F. is therefore, $C_{1}e^{-x}+C_{2}e^{7x}$.

To find P. I.

Let, $y_{p}=Axe^{-x}+Bx+C$

Then, $Dy_{p}=-Axe^{-x}+Ae^{-x}+B$

and $D^{2}y_{p}=Axe^{-x}-2Ae^{-x}$

Since $y_{p}$ is a root of the given differential equation,

$\quad D^{2}y_{p}-6Dy_{p}-7y_{p}=8e^{-x}-7x-6$

$\Rightarrow Axe^{-x}-2Ae^{-x}+6Axe^{-x}-6Ae^{-x}-6B-7Axe^{-x}-7Bx-7C=8e^{-x}-7x-6$

$\Rightarrow -8Ae^{-x}-7Bx-6B-7c=8e^{-x}-7x-6$

Equating like terms in both sides, we get

$\quad\quad -8A=8$

$\quad\quad -7B=-7$

$\quad\quad -6B-7C=-6$

$\implies$

$\quad\quad A=-1,\:B=1,\:C=0$

$\therefore y_{p}=-xe^{-x}+x$

$\therefore$ complete solution is

$\quad y=C_{1}e^{-x}+C_{2}e^{7x}-xe^{-x}+x$