75.9.2 problem 221

Internal problem ID [16768]
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 : Monday, March 31, 2025 at 03:16:40 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _dAlembert]

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 36

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

\[ y = x -\frac {x^{2}}{4}+\operatorname {LambertW}\left (\frac {c_1 \,{\mathrm e}^{\frac {x}{2}-1}}{2}\right )^{2}+2 \operatorname {LambertW}\left (\frac {c_1 \,{\mathrm e}^{\frac {x}{2}-1}}{2}\right )+1 \]

✓ Mathematica. Time used: 2.38 (sec). Leaf size: 177

ode=y[x]==x*(1+D[y[x],x])+D[y[x],x]^2; 
ic={}; 
DSolve[{ode,ic},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*}

✓ Sympy. Time used: 3.708 (sec). Leaf size: 75

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*(Derivative(y(x), x) + 1) + y(x) - Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ C_{1} + x + 2 \sqrt {\frac {x^{2}}{4} - x + y{\left (x \right )}} - 2 \log {\left (\sqrt {\frac {x^{2}}{4} - x + y{\left (x \right )}} + 1 \right )} = 0, \ C_{1} + x - 2 \sqrt {\frac {x^{2}}{4} - x + y{\left (x \right )}} - 2 \log {\left (\sqrt {\frac {x^{2}}{4} - x + y{\left (x \right )}} - 1 \right )} = 0\right ] \]