Example 5. Integrate the differential equation $y^{\prime}=\frac{x \sqrt{1+x^{2}}+1}{\sqrt{1+x^{2}}\left(y+e^{y}\right)}$ and also find the particular solution that satisfies the Cauchy condition: $x_{0}=2, y_{0}=1$.
Solution. We transform the given equation to separate the variables (replacing $y^{\prime}=\frac{d y}{d x}$):

$$
\frac{d y}{d x}=\frac{x \sqrt{1+x^{2}}+1}{\sqrt{1+x^{2}}\left(y+e^{y}\right)}, \quad\left(y+e^{y}\right) d y=\frac{x \sqrt{1+x^{2}}+1}{\sqrt{1+x^{2}}} \cdot d x
$$

We integrate the obtained equality: $\int\left(y+e^{y}\right) d y=\int\left(x+\frac{1}{\sqrt{1+x^{2}}}\right) d x$, $\frac{y^{2}}{2}+e^{y}=\frac{x^{2}}{2}+\ln \left(x+\sqrt{1+x^{2}}\right)+C$.

We have obtained the general solution in implicit form. To find the particular solution, we substitute the values $x=2, y=1$ into the general solution and find the constant $C$: $\frac{1}{2}+e=2+\ln (2+\sqrt{5})+C, C=e-$ $-\frac{3}{2}-\ln (2+\sqrt{5})$. Substituting this value of $C$ into the formula for the general solution, we obtain the desired particular solution in implicit form: $\frac{y^{2}}{2}+e^{y}=\frac{x^{2}}{2}+\ln \left(x+\sqrt{1+x^{2}}\right)+e-\frac{3}{2}-\ln (2+\sqrt{5})$.

Answer. $\quad \frac{y^{2}-x^{2}}{2}+e^{y}-\ln \left(x+\sqrt{1+x^{2}}\right)=C ; \quad \frac{y^{2}-x^{2}}{2}+e^{y}-$ $-\ln \frac{x+\sqrt{1+x^{2}}}{2+\sqrt{5}}=e-\frac{3}{2}$.

Sometimes, to bring the equation to a form convenient for integration, a non-trivial substitution (change of variables) can be used.