58.2.12 problem 13

Internal problem ID [9135]
Book : Second order enumerated odes
Section : section 2
Problem number : 13
Date solved : Sunday, March 30, 2025 at 02:23:06 PM
CAS classification : [_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} 10 y^{\prime \prime }+x^{2} y^{\prime }+\frac {3 {y^{\prime }}^{2}}{y}&=0 \end{align*}

✓ Maple. Time used: 0.021 (sec). Leaf size: 49

ode:=10*diff(diff(y(x),x),x)+x^2*diff(y(x),x)+3/y(x)*diff(y(x),x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {10 y^{{13}/{10}}}{13}-\frac {3 c_1 x \operatorname {WhittakerM}\left (\frac {1}{6}, \frac {2}{3}, \frac {x^{3}}{30}\right ) {\mathrm e}^{-\frac {x^{3}}{60}} 30^{{1}/{6}}}{4 \left (x^{3}\right )^{{1}/{6}}}-c_1 x \,{\mathrm e}^{-\frac {x^{3}}{30}}-c_2 = 0 \]

✓ Mathematica. Time used: 66.264 (sec). Leaf size: 73

ode=10*D[y[x],{x,2}]+x^2*D[y[x],x]+3/y[x]*(D[y[x],x])^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_2 \exp \left (\int _1^x\frac {30 e^{-\frac {1}{30} K[1]^3} \sqrt [3]{K[1]^3}}{30 c_1 \sqrt [3]{K[1]^3}-13 \sqrt [3]{30} \Gamma \left (\frac {1}{3},\frac {K[1]^3}{30}\right ) K[1]}dK[1]\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), x) + 10*Derivative(y(x), (x, 2)) + 3*Derivative(y(x), x)**2/y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE x**2*y(x)/6 - sqrt(-(-x**4*y(x) + 120*Derivative(y(x), (x, 2)))*y(x))/6 + Derivative(y(x), x) cannot be solved by the factorable group method