2.19   ODE No. 19

1. Problem in Latex
2. Mathematica input
3. Maple input

$$y^{\prime }(x)-(y(x)+x{)}^2=0$$ ✓ Mathematica : cpu = 0.0549314 (sec), leaf count = 14

$$\{\{y(x)\to -x+\tan (x+c_1)\}\}$$ ✓ Maple : cpu = 0.039 (sec), leaf count = 16

$$\{y\left(x\right)=-x-\tan (-x+\mathit{\_}\mathit{C}\mathit{1})\}$$

Hand solution

$$\begin{array}{rl}{y}^{\prime }-{(y+x)}^2& =0\\ \text{(1)}& y^{\prime }& ={(y+x)}^2\end{array}$$

This is Riccati first order non-linear ODE of the form. Let $u=y+x$, then $u^{\prime }=y^{\prime }+1$ and (1) becomes$$\begin{array}{rl}{u}^{\prime }-1& =u^2\\ u^{\prime }& =1+u^2\end{array}$$

This is separable$$\begin{array}{rl}\frac{du}{dx}\frac{1}{1+u^2}& =1\\ \int \frac{du}{1+u^2}& =\int dx\\ {\tan }^{-1}u& =x+C\\ u& =\tan (x+C)\end{array}$$

Since $u=y+x$ then

$$y=\tan (x+C)-x$$