Find the values of the constants a and b for which ax + b is a particular integral of the differential equation 2y' + 5y = 10x. Hence find the general solution of 2y' + 5y = 10x .

We start by letting yp = ax+b, as suggested, and finding the derivative yp' = a. Substituing into ODE  (by way of matching coefficients) gives a pair of simulataneous (algebraic) equations:

(1) 2a=5b

(2) -5a=10

which can be solved to give (a,b)=(-2,-4/5). Then yp=-2x-4/5.

We now find the characteristic solution to the homogeneous ODE, 2y'+5y=0. By rearranging and integrating we find that 

2ln|y| = 5x+C

which we rearrange to find yc = Aexp(5x/2). Then the general solution, given by y(gs)=yc+yp, takes the form

y=Aexp(5x/2)-2x-4/5,

and we are done.