60.1.480 problem 493

Internal problem ID [10494]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 493
Date solved : Sunday, March 30, 2025 at 05:10:21 PM
CAS classification : [`y=_G(x,y')`]

\begin{align*} \left (y^{2}-2 a x +a^{2}\right ) {y^{\prime }}^{2}+2 a y y^{\prime }+y^{2}&=0 \end{align*}

✓ Maple. Time used: 0.184 (sec). Leaf size: 113

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

\begin{align*} y &= 0 \\ \left [x \left (\textit {\_T} \right ) &= \frac {{\operatorname {arctanh}\left (\frac {1}{\sqrt {\textit {\_T}^{2}+1}}\right )}^{2} \sqrt {\textit {\_T}^{2}+1}\, a^{2}+\left (-2 a c_1 \sqrt {\textit {\_T}^{2}+1}-2 a^{2}\right ) \operatorname {arctanh}\left (\frac {1}{\sqrt {\textit {\_T}^{2}+1}}\right )+\left (c_1^{2}+a^{2}\right ) \sqrt {\textit {\_T}^{2}+1}+2 c_1 a}{2 \sqrt {\textit {\_T}^{2}+1}\, a}, y \left (\textit {\_T} \right ) = \frac {\left (-a \,\operatorname {arctanh}\left (\frac {1}{\sqrt {\textit {\_T}^{2}+1}}\right )+c_1 \right ) \textit {\_T}}{\sqrt {\textit {\_T}^{2}+1}}\right ] \\ \end{align*}

✓ Mathematica. Time used: 2.366 (sec). Leaf size: 149

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

\[ \text {Solve}\left [\left \{x=\frac {a^2 K[1]^2+2 a K[1] y(K[1])+K[1]^2 y(K[1])^2+y(K[1])^2}{2 a K[1]^2},y(x)=\exp \left (\int _1^{K[1]}\frac {1}{K[2] \left (K[2]^2+1\right )}dK[2]\right ) \int _1^{K[1]}\frac {a \exp \left (-\int _1^{K[3]}\frac {1}{K[2] \left (K[2]^2+1\right )}dK[2]\right )}{K[3]^2+1}dK[3]+c_1 \exp \left (\int _1^{K[1]}\frac {1}{K[2] \left (K[2]^2+1\right )}dK[2]\right )\right \},\{y(x),K[1]\}\right ] \]

✗ Sympy

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

Timed Out