Internal problem ID [5335]
Book: Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres.
McGraw Hill 1952
Section: Chapter 9. Equations of first order and higher degree. Supplemetary problems. Page
65
Problem number: 29.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _dAlembert]
\[ \boxed {y-\left (1+y^{\prime }\right ) x -{y^{\prime }}^{2}=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 39
dsolve(y(x)=(1+diff(y(x),x))*x+diff(y(x),x)^2,y(x), singsol=all)
\[ y = \left (\operatorname {LambertW}\left (\frac {c_{1} {\mathrm e}^{\frac {x}{2}-1}}{2}\right )-\frac {x}{2}+2\right ) x +{\left (\operatorname {LambertW}\left (\frac {c_{1} {\mathrm e}^{\frac {x}{2}-1}}{2}\right )-\frac {x}{2}+1\right )}^{2} \]
✓ Solution by Mathematica
Time used: 1.048 (sec). Leaf size: 177
DSolve[y[x]==(1+y'[x])*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*}