Differential Equation Question: 2nd Order DE. Need Help.

Question

Using the method of undetermined coefficients, determine the general solution of the following second-order, linear, non-homogenous equations.

 

y'' - 2y' + 2y = sin(x) + cos(2x)

Answer 1

Since the RHS is sine and cosine functions, I am going to guess that the actual solution will be of the form

 

y = Asinx + Bcosx + Csin(2x) + Dcos(2x).

 

I will take the first and second derivatives and plug them into the left side of the equation, and see if I can determine the coefficients A, B, C, and D that will give me the RHS of the equation.

 

y' = Acosx-Bsinx+2Ccos(2x)-2Dsin(2x)

y'' = -Asinx-Bcosx-4Csin(2x)-4Dcos(2x)

 

Plug all of this into the LHS and get

 

(A+2B)sinx + (B-2A)cosx + (4D-2C)sin(2x) + (-2D-4C)cos(2x)

 

Since the RHS has only a sinx term and a cos(2x) term, each with a coefficient of 1, we must make the coefficient of our sinx = 1, the coefficient of our cosx = 0, the coefficient of our sin(2x)=0, and the coefficient of our cos(2x) = 1

 

So you must solve:

A+2B=1

B-2A=0

4D-2C=0

-2D-4C=1

 

These will give the the coefficients you need to write the general solution for the diff eq.

 

(I got A = 1/5;  B=2/5;  C=-(1/5);    D=-(1/10)   -  I checked it and these worked, but you should do the work and arrive at these coefficients yourself...  :-)   )

 

 

Answer 2

We first find the general solution of the homogeneous  equation

y'' - 2y' + 2y =0. As usual, characteristic equation is k²-2k+2=0, that gives the two complex roots 1- i and its conjugate 1+i, So the general solution is y(x)=C₁e^x+C₂xe^x..

Now the only thing left is to find a particular solution of the inhomogeneous equation y"-2y'+2y=sinc(x)+cos(2x). Depending on RHS, we have to look for a a particular solution in the form A sin(x)+b cos(x)+C sin(2x)+ D cos(2x).

If we substitute it into the equation (which one), we find

A= 1/5, B= 2/5,C=-(3/16),D= -(1/8). That all. So, 

y_p(x)=(1/5) sin(x)+(2/5) cos(x)-(3/16) sin(2x)-(1/8) cos(2x).

The general solution is y_G(x)=y(x)-y_p(x).

Check the answer.