Riccati Equation

Rewriting the given equation, $\begin{aligned} \frac{dy}{dx}&=y+e^x+\frac{y^2}{e^x}\\ \frac{dy}{dx}&=y+e^x\left(1+\left(\frac{y}{e^x}\right)^2\right)\\ \frac{\frac{dy}{dx}-y}{e^x}&=\left(1+\left(\frac{y}{e^x}\right)^2\right) \end{aligned}$

Substituting $u=\frac{y}{e^x}$ , we get $\begin{aligned} \frac{du}{dx}&=\frac{\frac{dy}{dx}e^x-ye^x}{e^{2x}}\\ &=\frac{\frac{dy}{dx}-y}{e^{x}} \end{aligned}$

$\begin{aligned} \frac{du}{1+u^2}&=dx \\ \int\frac{du}{1+u^2}&=\int dx \\ \arctan u&=x+c\\ y(x)&=e^x\tan (x+c) \end{aligned}$

So, $y(\pi /4)$ when $c=0,$ is $e^{\pi /4}\tan(\pi /4)\approx 2.193.$