need help solving this

Dear Ames,

With c(t)=0 we've got a very simple solution:
a(t)=k1*exp(-p1*t) , b(t)=k2*exp(-p2*t) k1, k2 real constants (1) .
for c(t) # 0 trying an additive form:
a(t)=k1*exp(-p1*t)+ m(t) ; b(t)=k2*exp(-p2*t)+ r(t)
bring us into 'a loop' :
m'(t)= -p1*m(t) -p3*c(t) ; r'(t)=p3*c(t)-p2*r(t) (2)
we recognize in (2) our initial equation ( a->m ,b->r ),

So we must use a different method named " constants variation " it means
in (1) k1->k1(t) k2->k2(t) trying this way to generalize sol.(1) ,
you compute a'(t) and b'(t) , a'(t)= d/dt ( k1(t)*exp(-p1*t))= ...
We obtain after some computing:
k1(t)=int(-p3*c(t)*exp(-p1*t) )+ c1 ,...and a(t),b(t),
observe c(t) might be any continuous real function,
ALAIN.
.