this is the sort of thing you should look up in your text book.

for second order DEs:

Homogeneous means it looks like ......the 0 is what makes it homogeneous, in general, there would be a non-zero function on the right hand side.

Linear means we have a differential equation of the form in which the variables are the y-components, and all these components stand alone. meaning you only have things like y, y', etc and not things like yy' or y'y'' as the variables.

for the first one (we need the method of undetermined coefficients), see here. for the second one, see hereThen I must find the general solution for the second and the particular solution with dy/dx=0, y=1 when x=0

hope that helps

it would be a good idea to look up linear and non-linear ODEs, homogeneous ODEs, non-homogeneous ODEs and second-order ODEs in your textbook. just look those up in the index. short of doing the problems for you (the majority of the second problem is done in one of the links i gave you, i think) it would take a lot of re-inventing the wheel to explain this. read your text and if you have specific questions, we can help