Question

can someone please help me understand undetermined coefficients

Answer 1

well if you have a non-homogeneous linear differential equation then to find the general solution you add the complimentary function to a particular solution. The complementary solution you simply set the left hand side of the equation for zero, and solve like you would for a homogeneous DE.

Undetermined coefficients is a groups of methods you can use to find a particular solution of the DE. It is usually good to start by assuming that this equation will have the same form as the right hand side of the DE.

The most common approach to finding this is the superposition method which i'll quickly explain

For example if you have \[y''+4y'-2y=2x^{2}-3x+6 \]
then you can start by guessing that a particular solution will also be a quadratic polynomial. So you just need to write down\[y_p=Ax^{2}+Bx+C\]
and then differentiate it
\[y_p'=2Ax+b\]
\[y_p''=2A\]

so then you just plug those values of y,y' and y'' it into the left hand side of the original equation which gives you\[2A+4(2Ax+B)-2(Ax^{2}+Bx+C)=2x^{2}-3x+6\]
then group the like terms. (This is kind of like what you do when you use partial fractions)
\[(-2Ax^{2)}+(8Ax-2Bx)+(2A+4B-2C)=2x^2-3x+6\]
\[(-2A)x^{2}+(8A-2B)x+(2A+4B-2C)=2x^2-3x+6\]
from there you have three systems of equations to solve. Look on the right side, and see how the x^2 is multiplied by two. Now look on the left and see how it is multiplied by -2A. So you just say \[-2A=2\]
and similarly\[8A-2B=-3\] \[2A+4B-2C=6\]

solving gives you the values of A,B and C you need to get a particular solution. Which you add to the complimentary solution to get a general solution.

Now one thing you have to be careful about is that your particular solution does not have any terms that look like terms in your complimentary solution. for example if your complimentary system has Ae^x in it, then your particular solution can't. If this is the case, then usually you can solve this by multiplying your "guess" solution by x^n. Where n is the smallest integer that eliminates the problem.

hope that makes sense