75.12.39 problem 313

Internal problem ID [16826]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 12. Miscellaneous problems. Exercises page 93
Problem number : 313
Date solved : Thursday, March 13, 2025 at 08:52:33 AM
CAS classification : [_rational, _dAlembert]

\begin{align*} y^{\prime }+x {y^{\prime }}^{2}-y&=0 \end{align*}

✓ Maple. Time used: 0.104 (sec). Leaf size: 59

ode:=diff(y(x),x)+diff(y(x),x)^2*x-y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = 2 x \,{\mathrm e}^{\operatorname {RootOf}\left (-x \,{\mathrm e}^{2 \textit {\_Z}}+2 x \,{\mathrm e}^{\textit {\_Z}}+\textit {\_Z} +c_{1} -x -{\mathrm e}^{\textit {\_Z}}\right )}+\operatorname {RootOf}\left (-x \,{\mathrm e}^{2 \textit {\_Z}}+2 x \,{\mathrm e}^{\textit {\_Z}}+\textit {\_Z} +c_{1} -x -{\mathrm e}^{\textit {\_Z}}\right )+c_{1} -x \]

✓ Mathematica. Time used: 0.856 (sec). Leaf size: 46

ode=D[y[x],x]+x*D[y[x],x]^2-y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\left \{x=\frac {\log (K[1])-K[1]}{(K[1]-1)^2}+\frac {c_1}{(K[1]-1)^2},y(x)=x K[1]^2+K[1]\right \},\{y(x),K[1]\}\right ] \]

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (sqrt(4*x*y(x) + 1) - 1)/(2*x) cannot be solved by the factorable group method