# Solution of a nonhomogeneous equation

• median27
In summary, the person is trying to solve for y, but they get stuck because they don't know how to do the derivatives. They ask for help, and the person responds by telling them how to do the derivatives and telling them that y=C1e^-x+C2e^-2x+x^2.
median27
(i d0nt kn0w how to use LaTeX)

1)D^2(D-1)y=3e^x+sinx

for yc:
let y=e^mx
D^2(D-1)e^mx=0
m^2(m-1)e^mx=0
f(m)=0
m^2(m-1)=0
m=0,0,1

yc= C1+C2x+C3e^x

for yp:
R(x)=3e^x+sinx
m'=1,+/- 1i

yp=Axe^x+Bcosx+Csinx
yp'=A(xe^x+e^x)-Bsinx+Ccosx
yp"=A(xe^x+2e^x)-Bcosx-Csinx

D^2(D-1)yp=3e^x+sinx

Guys can you help me find the value of A, B and C. I had a hard time simplifying the terms and can't equate the coefficients. Thanks for analyzing my post this far. :D

You're doing fine so far. You need to differentiate one more time because

D2(D-1)yp = (D3-D2)yp=yp'''-yp''

Last edited:
Thanks for your help vela! I simply overlooked the problem and didn't get the third derivative. :D

Here's how I would have done it:
If $D^@(D-1)y= 3e^x+ sin(x)$ then $D(D-1)y= 3e^x- cos(x)+ C$ and $(D-1)y= 3e^x- sin(x)+ Cx+ D$

Then solve the characteristic equation m- 1= 0 to get m= 1 which means that $e^x$ is a solution to the associated homogeneous equation. Since part of the right side is $e^x$ look for a solution of the form $Axe^x+ Bsin(x)+ Ccos(x)+ Ex+ F$.

Can you figure out my mistake. The answer is supposed to be:
y=C1e^-x+C2e^-2x+6x^2-18x+21

Here is the problem and my solution:

2) (D^2+3D+2)y=12x^2

for yc
let y=e^mx
(D^2+3D+2)e^mx=0
(m^2+3m+2)e^mx=0
f(m)=0
m^2+3m+2=0
m=-1,-2

yc=C1e^-x+C2e^-2x

for yp
R(x)=12x^2
m'=0,0,0
yp=A+Bx+Cx^2
yp'=B+2Cx^2
yp"=4Cx^2

(D^2+3D+2)yp=12x^2

4Cx^2+3B+6Cx^2+2A+2Bx+2Cx^2=12x^2

3B+12Cx^2+2A+2Bx=12x^2

Equating coefficients
i get C=1, B&A=0

Giving:
yp=x^2

y=C1e^-x+C2e^-2x+x^2
(which is wr0ng)

Thanks for any help!

P.s. An0ther problem was posted.

You didn't calculate the derivatives of yp correctly.

Alright, i get it. :D I'm used to differentiate e^n terms and overlooked x^n terms. I'm becoming reckless. Thanks vela.

## What is a nonhomogeneous equation?

A nonhomogeneous equation is an equation where the terms do not have the same degree or order. This means that the equation cannot be solved by using algebraic methods alone. Instead, it requires the use of more advanced techniques such as integration or differential equations.

## How is a nonhomogeneous equation solved?

A nonhomogeneous equation is solved by using a combination of algebraic and mathematical methods. This typically involves breaking down the equation into smaller, more manageable parts and then using techniques such as integration, differentiation, or substitution to solve for the unknown variable.

## What is the role of the nonhomogeneous term in the equation?

The nonhomogeneous term in an equation is the part that does not have the same degree or order as the rest of the terms. It typically represents a source or input in the equation and can affect the overall solution. In some cases, the nonhomogeneous term may be equal to zero, making the equation homogeneous and easier to solve.

## Can a nonhomogeneous equation have more than one solution?

Yes, a nonhomogeneous equation can have multiple solutions. This is because the equation may have different initial conditions or boundary conditions that result in different solutions. Additionally, some nonhomogeneous equations may have infinite solutions or no solutions at all.

## What are some real-life applications of nonhomogeneous equations?

Nonhomogeneous equations have various applications in real life, including in physics, engineering, and economics. For example, they can be used to model systems that involve external forces, such as a spring-mass system with friction. They are also commonly used in differential equations to model physical processes such as heat transfer and population growth.

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