29.37.11 problem 1129

Internal problem ID [5669]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 37
Problem number : 1129
Date solved : Sunday, March 30, 2025 at 09:55:38 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 1.664 (sec). Leaf size: 134

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

\begin{align*} y &= \sqrt {-a}\, x \\ y &= -\sqrt {-a}\, x \\ y &= -\frac {\left (a^{2} \left (a -1\right ) x^{-\frac {\sqrt {\left (a -1\right ) a}}{a}}-x^{\frac {\sqrt {\left (a -1\right ) a}}{a}} c_1^{2}\right ) x}{2 \sqrt {\left (a -1\right ) a}\, c_1} \\ y &= -\frac {x \left (-x^{-\frac {\sqrt {\left (a -1\right ) a}}{a}} c_1^{2}+x^{\frac {\sqrt {\left (a -1\right ) a}}{a}} a^{2} \left (a -1\right )\right )}{2 \sqrt {\left (a -1\right ) a}\, c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.444 (sec). Leaf size: 113

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

\begin{align*} y(x)\to \frac {1}{2} e^{-c_1} x^{1-\sqrt {\frac {a-1}{a}}} \left (-a x^{2 \sqrt {\frac {a-1}{a}}}+e^{2 c_1}\right ) \\ y(x)\to \frac {1}{2} e^{c_1} x^{\sqrt {\frac {a-1}{a}}+1}-\frac {1}{2} a e^{-c_1} x^{1-\sqrt {\frac {a-1}{a}}} \\ \end{align*}

✓ Sympy. Time used: 20.029 (sec). Leaf size: 673

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

\[ \text {Solution too large to show} \]