As to (1), what did you get as your specific solution to the non-homogeneous equation (to which you added the general homogeneous solution)?
I've been attempting these two questions a few times and I countinue to get them wrong.
1) Find y as a function of x if
So I solved the Homogenous equation as a Euler equation. My homogeneous solution is yh= C1x^(8)+C2ln(x)x^(8)
then i proceeded with undetermined coefficients and ended up with 151/25x^8-1336/25ln((x))x^8+x^(3)/25
Whats wrong with the method?
2) Use the method of undetermined coefficients or the method of differential operators to find one solution of
y'' -8y' + 43y = 32e^4t(cos(5t))+64e^4t(sin(5t))+ 7e^0t
(It doesn't matter which specific solution you find for this problem.)
My guess for yp was Ae^4t cos(5t)+Be^ 4t sin(5t)
And I used the undetermined coefficent method to find A=16 and B=32.
Your specific solution seems to be correct, as does your general homgeneous solution. Thus, you have general solution .
Setting y(1)=6, we get
you have , which by the above, you can see is incorrect.
With , set y'(1)=-5, and solve for the correct .