Since, the curve passes through origin.
Thus, equation 2 becomes
1 = C
Substituting C = 1 in equation 2, we get,
\[x\text{ }+\text{ }y\text{ }+\text{ }1\text{ }=\text{ }{{e}^{x}}\]
Therefore, the required general solution of the given differential equation is
\[x\text{ }+\text{ }y\text{ }+\text{ }1\text{ }=\text{ }{{e}^{x}}\]