This is just an exercise is plug and chug.
Now lets compute some derivatives!
Now just plug into the ODE to get
Since u_1 and u_2 satisfy the homogenous equation the first two terms are zero.
How does one show that v(x) solves the equation Lv=f
where L is the differential operator: Lu= -(pu')'+qu=f
and v(x)= -(int from a to x) f(y)*u1(y)*u2(x)/(p(y)*W(y)) dy - (int from x to b) f(y)*u1(x)*u2(y)/(p(y)*W(y)) dy
where W(y) is the Wronskian