put the y terms on the left
y''+y'=2x
find the complementary function
y''+y'=0
form the quadratic
k^2 + k = 0
solve it
k = -1 or 0
the complementary function is:
y = Ae^(-x) + B
Find the particular integral
P(x)= ax^2 + bx + c
P'(x)= ax + b
P''(x)= a
P''(x) + P'(x) = 2x
a + ax + b = 2x
a = 2 b = -2
P(x)= 2x^2 - 2x
Add the complementary function and the particular integral to give the general solution
y(x)= Ae^(-x) + 2x^2 - 2x + B
Differentiate
y'(x)= -Ae^(-x) + 2/3 x^3 -2
plug in the initial values to find A and B
A + B = 0
-A -2 = 1
A = -3
B = 3
And voila
y(x)= -3e^-x + 2x^2 - 2x + 3
Please note that I didn't stop to check my answer so plug it in and see if it fits.
That was probably fairly confusing but these are actually quite easy. I started studying these, amongst other things, on Thursday and on Monday I have an exam, so I hope I understand them.