54.7.79 problem 1688 (book 6.97)

Internal problem ID [12928]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1688 (book 6.97)
Date solved : Friday, October 03, 2025 at 03:50:50 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} x^{4} y^{\prime \prime }-x \left (x^{2}+2 y\right ) y^{\prime }+4 y^{2}&=0 \end{align*}

✓ Maple. Time used: 0.026 (sec). Leaf size: 22

ode:=x^4*diff(diff(y(x),x),x)-x*(x^2+2*y(x))*diff(y(x),x)+4*y(x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = x^{2} \left (-c_1 \tanh \left (c_1 \left (\ln \left (x \right )-c_2 \right )\right )+1\right ) \]

✓ Mathematica. Time used: 0.146 (sec). Leaf size: 178

ode=4*y[x]^2 - x*(x^2 + 2*y[x])*D[y[x],x] + x^4*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\frac {x^2}{-c_1 x^4+2 K[2] x^2-K[2]^2}dK[2]-\int _1^x\left (\frac {\left (\frac {y(x)^2}{K[3]^4}+c_1\right ) K[3]^3}{-c_1 K[3]^4+2 y(x) K[3]^2-y(x)^2}+\int _1^{y(x)}\left (\frac {2 K[3]}{-c_1 K[3]^4+2 K[2] K[3]^2-K[2]^2}-\frac {K[3]^2 \left (4 K[2] K[3]-4 c_1 K[3]^3\right )}{\left (-c_1 K[3]^4+2 K[2] K[3]^2-K[2]^2\right ){}^2}\right )dK[2]\right )dK[3]=c_2,y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**4*Derivative(y(x), (x, 2)) - x*(x**2 + 2*y(x))*Derivative(y(x), x) + 4*y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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