Question

Solve the differential equation:

d2y/dx2 - 2 dy/dx + y = 3sinhx

The answer should be:
y(x) = e^x (Ax+ B) + ( 3^8 )(2[x^2][e^x] - [e^-x] )

Can someone please show me how to work it out?

Answer 1

There is probably a mistake in the transcription of the solution:
y(x) = e^x (Ax+ B) + ( 3<b>/</b>8 )(2[x^2][e^x] - [e^-x] )
Namely the power 3^8 should be a division as shown in bold.

Given:
y"-2y'+y=3sinh(x) .....(0)

This is a linear (y and its derivatives appear in first power) second order (maximum y") non-homogeneous (right-hand side is not zero) differential equation with constant coefficients (coefficients of y and its derivatives are constants).

The general solution of the homogeneous equation with constant coefficients is a combination of terms A_ix^je^k where i,j,k could be complex.

We will first solve the homogeneous equation, namely drop the right hand side to get:
y"-2y'+y=0 .....(1)
From (1), we form and solve the characteristic polynomial in z which has the same coefficients as (1).
z²-2z+1=0......(2)
to get
(z-1)(z-1)=0
or z=1 (multiplicity 2).
The general solution of the homogeneous equation is a linear combination of the basis formed by the solutions of the characteristic equation (2):
A_ie^zi.
This applies when the solutions of zi are distinct. If multiplicity occurs, as in this case, terms will contain successive powers of x.
Therefore with a multiplicity of 2 for z=1, the solution to the homogeneous equation is:
k₁xe^x+k₂e^x
where the power of e is 1, equivalent to the solution z=1.
To determine the solution of the non-homogeneous equation, we will find a particular solution.

Since the right hand side is 3sinh(x)=(3/2)(e^x-e^-x),
we can use the method of undetermined coefficients.
Since one of the terms on the right-hand-side (e^x) corresponds to one of the terms in the solution basis, we need to multiply the term by powers of x until it is distinct from any term of the general solution.
The assumed particular will then take the form:
yp(x) = Ax²e^x+Be^-x

The next step is to find the coefficients of A and B.
This can be done by substituting y(p) into equation (0), carry out the differentiations and compare coefficients of e^x and e^-x to get
A=3/4, and B=-3/8

The solution to the differential equation (0) is therefore the sum of the general solution and the particular solution:

y=(3/4)x²e^x - (3/8)e^-x + (k₁x+k₂)e^x

where k₁ and k₂ are integration constants.

The above expression can be rearranged to be identical with the given answer.


References for further reading:
Differential Equations by Richard Bronson & Gabriel Costa, Shaum's Outline Series, McGraw-Hill.
http://en.wikipedia.org/wiki/Linear_differential_equation
http://en.wikipedia.org/wiki/Differential_equation

Answer 2

what is the answer to this problem

6(6g-2)+8(1-5g)=2g