75.9.2 problem 221

Internal problem ID [16839]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 8.3. The Lagrange and Clairaut equations. Exercises page 72
Problem number : 221
Date solved : Tuesday, January 28, 2025 at 09:34:33 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _dAlembert]

\begin{align*} y&=x \left (1+y^{\prime }\right )+{y^{\prime }}^{2} \end{align*}

✓ Solution by Maple

Time used: 0.088 (sec). Leaf size: 36

dsolve(y(x)=x*(1+diff(y(x),x))+diff(y(x),x)^2,y(x), singsol=all)
 

\[ y = x -\frac {x^{2}}{4}+\operatorname {LambertW}\left (\frac {c_{1} {\mathrm e}^{\frac {x}{2}-1}}{2}\right )^{2}+2 \operatorname {LambertW}\left (\frac {c_{1} {\mathrm e}^{\frac {x}{2}-1}}{2}\right )+1 \]

✓ Solution by Mathematica

Time used: 2.380 (sec). Leaf size: 177

DSolve[y[x]==x*(1+D[y[x],x])+D[y[x],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*}