Question: How to find a certain function?

woud like to find H(x)

just for one of definition H(x) + H(y) = H(x+y), but it seems require to duplicate one more set of differential system

Error, (in dsolve) found differentiated functions with same name but depending on different arguments in the given DE system: [{a(x), a(y)}, {b(x), b(y)}]

eq2a := a(x)*(diff(c(x), x))+b(x);
eq3a := a(x)*(diff(b(x), x))+c(x)*(diff(b(x), x));
eq4a := a(x)*(diff(c(x), x))+a(x)*(diff(b(x), x))+b(x);
eq2b := a(y)*(diff(c(y), y))+b(y);
eq3b := a(y)*(diff(b(y), y))+c(y)*(diff(b(y), y));
eq4b := a(y)*(diff(c(y), y))+a(y)*(diff(b(y), y))+b(y);
inis := diff(a(x),x)=H(a(x)), diff(b(x),x)=H(b(x)), diff(c(x),x)=H(c(x)), a(1) = b(1), b(1) = c(1), a(0) = b(0), b(0) = c(0);
inis := diff(a(x),x)=H(a(x)), diff(b(x),x)=H(b(x)), diff(c(x),x)=H(c(x));
inio := diff(a(y),y)=H(a(y)), diff(b(y),y)=H(b(y)), diff(c(y),y)=H(c(y)),H(a(x)) + H(b(x)) = H(a(x)+b(x));
sol := dsolve([eq2a=1, eq3a=exp(a), eq4a=0, eq2b=1, eq3b=exp(b), eq4b=0, inis, inio], [a(x), b(x), c(x), a(y), b(y), c(y), H(a(x)), H(b(x)), H(a(y)), H(b(y))]);