Internal problem ID [12161]
Book: DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR
PUBLISHERS, Moscow 1969.
Section: Chapter 8. Differential equations. Exercises page 595
Problem number: 91.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _dAlembert]
\[ \boxed {y-x \left (1+y^{\prime }\right )-{y^{\prime }}^{2}=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 39
dsolve(y(x)=x*(1+diff(y(x),x))+diff(y(x),x)^2,y(x), singsol=all)
\[ y \left (x \right ) = x \left (2+\operatorname {LambertW}\left (\frac {c_{1} {\mathrm e}^{\frac {x}{2}-1}}{2}\right )-\frac {x}{2}\right )+{\left (\operatorname {LambertW}\left (\frac {c_{1} {\mathrm e}^{\frac {x}{2}-1}}{2}\right )-\frac {x}{2}+1\right )}^{2} \]
✓ Solution by Mathematica
Time used: 3.313 (sec). Leaf size: 177
DSolve[y[x]==x*(1+y'[x])+(y'[x])^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [-\sqrt {x^2+4 y(x)-4 x}+2 \log \left (\sqrt {x^2+4 y(x)-4 x}-x+2\right )-2 \log \left (-x \sqrt {x^2+4 y(x)-4 x}+x^2+4 y(x)-2 x-4\right )+x=c_1,y(x)\right ] \text {Solve}\left [-4 \text {arctanh}\left (\frac {(x-5) \sqrt {x^2+4 y(x)-4 x}-x^2-4 y(x)+7 x-6}{(x-3) \sqrt {x^2+4 y(x)-4 x}-x^2-4 y(x)+5 x-2}\right )+\sqrt {x^2+4 y(x)-4 x}+x=c_1,y(x)\right ] \end{align*}