How to check if my string input followed the correct syntax in C++?

I'm new to this site and this is my first time to ask here.

My problem is I want to check if my string follows a correct pattern or syntax. I'm doing it with C++ String (std::string). I have already done this using C-Style string, however, I want to do it this time in C++ String. Sample problem below:

Input: 2y'' + 3y' - 2y = 0 or y'' = 4y

I want to check if the derivative input is in correct syntax like (a)y'' + (b)y' + (c)y = 0, a second order homogeneous equation. However, I still want to input a non-standard form equation like the second sample input that can be transposed and make it to standard form.

What I did before with it is remove all the white spaces, loop the entire string and check every index. Eg. if 'y' is found the next char should be '\'' or an arithmetic symbol like '-' or '+' or '=' then if it does not match, then, it must return false.

Or maybe I am just implementing this wrong. I'm new to programming and just taking a computer science course. Note: Sorry for my bad English and sorry if I did not written my code here. Its just way too long.