60.1.547 problem 560

Internal problem ID [10561]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 560
Date solved : Wednesday, March 05, 2025 at 12:05:56 PM
CAS classification : [`y=_G(x,y')`]

\begin{align*} a y \sqrt {{y^{\prime }}^{2}+1}-2 x y y^{\prime }+y^{2}-x^{2}&=0 \end{align*}

✓ Maple. Time used: 1.648 (sec). Leaf size: 1086

ode:=a*y(x)*(1+diff(y(x),x)^2)^(1/2)-2*x*y(x)*diff(y(x),x)+y(x)^2-x^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 45.126 (sec). Leaf size: 314

ode=-x^2 + y[x]^2 - 2*x*y[x]*D[y[x],x] + a*y[x]*Sqrt[1 + D[y[x],x]^2]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solve}\left [\int _1^{y(x)}\left (\frac {4 x K[1]}{\left (x^2+K[1]^2\right )^2}-2 \int \left (-\frac {2 K[1]}{\left (x^2+K[1]^2\right )^2}-\frac {4 \left (x^2-K[1]^2\right ) K[1]}{\left (x^2+K[1]^2\right )^3}\right ) \, dx\right )dK[1]+2 \left (\int \frac {x^2-y(x)^2}{\left (x^2+y(x)^2\right )^2} \, dx-\frac {\sqrt {a^2 y(x)^2 \left (-a^2 y(x)^2+x^4+2 x^2 y(x)^2+y(x)^4\right )}}{a^2 y(x) \left (x^2+y(x)^2\right )}\right )&=c_1,y(x)\right ] \\ \text {Solve}\left [\int _1^{y(x)}\left (\frac {4 x K[2]}{\left (x^2+K[2]^2\right )^2}-2 \int \left (-\frac {2 K[2]}{\left (x^2+K[2]^2\right )^2}-\frac {4 \left (x^2-K[2]^2\right ) K[2]}{\left (x^2+K[2]^2\right )^3}\right ) \, dx\right )dK[2]+2 \left (\frac {\sqrt {a^2 y(x)^2 \left (-a^2 y(x)^2+x^4+2 x^2 y(x)^2+y(x)^4\right )}}{a^2 y(x) \left (x^2+y(x)^2\right )}+\int \frac {x^2-y(x)^2}{\left (x^2+y(x)^2\right )^2} \, dx\right )&=c_1,y(x)\right ] \\ \end{align*}

✗ Sympy

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

Timed Out