23.4.111 problem 111

Internal problem ID [6413]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 111
Date solved : Friday, October 03, 2025 at 02:05:41 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 }&=-4 y^{2}+x \left (x^{2}+2 y\right ) y^{\prime } \end{align*}

✓ Maple. Time used: 0.014 (sec). Leaf size: 22

ode:=x^4*diff(diff(y(x),x),x) = -4*y(x)^2+x*(x^2+2*y(x))*diff(y(x),x); 
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: 63.515 (sec). Leaf size: 150

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

\begin{align*} y(x)&\to \frac {x^2 \left (\left (1-i \sqrt {-1-c_1}\right ) x^{2 i \sqrt {-1-c_1}}+\left (1+i \sqrt {-1-c_1}\right ) c_2\right )}{x^{2 i \sqrt {-1-c_1}}+c_2}\\ y(x)&\to \left (1+i \sqrt {-1-c_1}\right ) x^2\\ y(x)&\to \left (1+i \sqrt {-1-c_1}\right ) x^2\\ y(x)&\to \frac {2 x^2}{1+c_2 x^2} \end{align*}

✗ 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