This is of the form
Hence
This is second order linear ODE with varying coefficient. Solved using power series method
giving solutions using special functions (Bessel functions). Let
But
Hence from
Verification
ode:=2*diff(y(x),x)-3*y(x)^2-4*a*y(x)=b+c*exp(-2*a*x); uode:=diff(u(x),x$2)-2*a*diff(u(x),x)+3/4*(b+c*(exp(-2*a*x)))*u(x)=0; uSol:=dsolve(uode,u(x)); my_sol:=(-2/3)*diff(rhs(uSol),x)/rhs(uSol); odetest(y(x)=my_sol,ode); 0