Hi I am new to cfd and just apreciate. If somebody tell me how to solve this differential equation anlytically. dy/dx=ax+y y=0 at x=0 Thanks cfd-user

Assuming a is constant, the solution is

y = a * (e^x - x - 1)

It is obtained as follows:

1. Find the homogenous solution, Yh, to the homogenous equation, i.e., dy/dx - y = 0. This is readily solved to yield Yh = e^x.

2. Guess a private solution, Yp, that satistfy the original equation. Since the inhomogenious term is a 1st degree polynomial, it is reasonable to guess Yp has also such a form, Yp=Ax+B. Substitution reveals A=B=-a.

3. The solution, y, is made of the linear combination y = C*Yh + Yp, where the constant C is found by satisfying the BC y(x=0)=0. Substitution and solving yields C=a.

You now have the above solution that satisfies the ODE and its BC (verify!). This is a general method for solving linear ODEs. It works for simple inhomogenous terms and BC. For other cases and better explanation on ODE solution refer to any basic calculus textbook.