How Do You Solve Complex Differential Equations?

[relskid]

Homework Statement

Find the general solution to the differential equation:

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

Homework Equations

The Attempt at a Solution

r - 1 = 0

r = 1

y1 = c1e^x

i don't really understand how to pick the yp... do you just guess? i tried both Asinx + Bcos(2x) and Acosx -2Bsin(2x), and neither really worked out... but i may be doing it all wrong. my teacher went over it quickly, and our book doesn't cover it.

Homework Statement

show by means of the wronskian, that the second order differential equation y" + a1(x)y' + a0(x)y = 0 cannot have three linearly independent solutions y1, y2, y3.

The Attempt at a Solution

i have no attempt on this problem, because i have no idea what to do. i know how to do the wronskian (just the determinant), but that proves absolutely nothing.


thank you for any help!

 

[Defennder]

For the first one, the method of undetermined coefficients you might want to split up the DE into two: y' - y = sin x and y' - y = cos(2x). Find the particular solutions for these two (perhaps by the method of complex function coefficients) and then add them up for the particular solution of the original DE.

You need to understand what the Wronskian means, and not just how to compute it. What does the Wronskian of say n functions say about their linear dependence?

 

[relskid]

so for number one:

yp1 = Asinx + Bcosx
y'p1 = Acosx - Bcosx

yp2 = Acos(2x) - 2Bsin(2x)
y'p2 = -2Asin(2x) - 4Bcos(2x) ?and number two, when n functions are linearly dependent, the determinant equals zero, and if it is linearly independent, it does not equal zero. how do i prove that it equals zero? do i let y1, y2, or y3 be equal to anything?

 

[Defennder]

For 1, yes you're on the right track. Just figure out those coefficients.

For 2, susbstitute y1,y2,y3 into the DE: y_1'' + ay_1' + by_1 = 0. Do the same for y2,y3. Now you have a system of 3 linear equations which can be represented as a matrix equation Au=0. What does the fact that the Wronskian is non-zero (assume this to be true so you can disprove it later) say about the set of solutions to this homogenous matrix equation?

 

[relskid]

wow, ok, thanks. i worked the problem for those two yp, separated, and i got the write answer. and the second one, i did the substitution, and again, found it linearly dependent. THANK YOU SO MUCH! i had been stuck on these two problems over the entire weekend. thank you for not giving away the answer, but helping me find out on my own.