I am absolutely not familiar with differential equations. However, I am facing the following differential equation:

\begin{equation} a(x)y^{\prime}(x)+b(x)y(x)=c(x)\sqrt{y^{2}(x)+d(x)} \end{equation}

or written differently \begin{equation} y^{\prime}(x) = \tilde{p}(x)y(x)+ \tilde{q}(x)\sqrt{y^{2}(x)+ \tilde{r}(x)} \end{equation}

If it helps, for simplification one can rewrite using some assumptions on $\tilde{p}(x)$ and $\tilde{q}(x)$ \begin{equation} y^{\prime}(x) = \frac{1}{|h(x)|^{2}}\left(h(x)h^{\prime}(x)y(x)+q(x)\sqrt{y^{2}(x)+|h(x)|^{2}}\right) \end{equation}

I checked Zwillinger's handbook on differential equations and Handbook of exact solutions for ordinary differential equations by Andrei Polyanin and Valentin Zaitsev in the hope to find differential equations having a similar form. However, I could not find anything similar.

I tried to solve this equation using MATLAB using the following code:

```
syms x p(x) q(x) r(x) y(x)
ySq = y(x)^2;
rSq = r(x)^2;
part1 = q(x)*sqrt(ySq + rSq);
part2 = p(x)*y(x);
ode = (diff(y,x) == part1 + part2)
ySol(x) = dsolve(ode)
```

however, this did not result in a solution. I do not know how to proceed. Can you help me making the next steps or pointing me even to the solution? Thank you very much.

EDIT: Is it of any help if the differential equation is in one of the following forms? \begin{align} y^{\prime}(x) &= f(x)y(x)+g(x)\sqrt{y^{2}(x)+1}\\ y^{\prime}(x) &= g(x)\left(\frac{f(x)}{g(x)}y(x)+\sqrt{y^{2}(x)+1}\right) \end{align}