54.1.476 problem 489

Internal problem ID [11790]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 489
Date solved : Sunday, October 12, 2025 at 01:52:13 AM
CAS classification : [_rational]

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

✓ Maple. Time used: 1.904 (sec). Leaf size: 359

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

\begin{align*} y &= -\frac {2 \sqrt {\left (\left (a +1\right )^{2} \left (a x -\frac {1}{2} b +x \right )^{2} a \operatorname {RootOf}\left (b \int _{}^{\textit {\_Z}}-\frac {4 \textit {\_a} \,a^{2}-\sqrt {-{\mathrm e}^{\frac {4 a}{b}} {\mathrm e}^{\frac {4}{b}} \left (4 \textit {\_a} \,a^{3}+8 \textit {\_a} \,a^{2}+4 a \textit {\_a} -1\right )}\, {\mathrm e}^{-\frac {2 \left (a +1\right )}{b}}+8 a \textit {\_a} +4 \textit {\_a} +1}{\textit {\_a} \left (4 \textit {\_a} \,a^{2}+8 a \textit {\_a} +4 \textit {\_a} +a +2\right )}d \textit {\_a} -2 b \ln \left (2 a x -b +2 x \right )+4 c_1 a +4 c_1 \right )-\frac {\left (b x +c \right ) a^{2}}{4}+\frac {\left (-\frac {b x}{2}-c \right ) a}{2}-\frac {b^{2}}{16}-\frac {c}{4}\right ) a}}{a \left (a +1\right )} \\ y &= \frac {2 \sqrt {\left (\left (a +1\right )^{2} \left (a x -\frac {1}{2} b +x \right )^{2} a \operatorname {RootOf}\left (b \int _{}^{\textit {\_Z}}-\frac {4 \textit {\_a} \,a^{2}-\sqrt {-{\mathrm e}^{\frac {4 a}{b}} {\mathrm e}^{\frac {4}{b}} \left (4 \textit {\_a} \,a^{3}+8 \textit {\_a} \,a^{2}+4 a \textit {\_a} -1\right )}\, {\mathrm e}^{-\frac {2 \left (a +1\right )}{b}}+8 a \textit {\_a} +4 \textit {\_a} +1}{\textit {\_a} \left (4 \textit {\_a} \,a^{2}+8 a \textit {\_a} +4 \textit {\_a} +a +2\right )}d \textit {\_a} -2 b \ln \left (2 a x -b +2 x \right )+4 c_1 a +4 c_1 \right )-\frac {\left (b x +c \right ) a^{2}}{4}+\frac {\left (-\frac {b x}{2}-c \right ) a}{2}-\frac {b^{2}}{16}-\frac {c}{4}\right ) a}}{a \left (a +1\right )} \\ \end{align*}

✓ Mathematica. Time used: 45.533 (sec). Leaf size: 27003

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

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✗ Sympy

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

Timed Out