83.48.20 problem Ex 21 page 116

Internal problem ID [19533]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Book Solved Excercises. Chapter VII. Exact differential equations.
Problem number : Ex 21 page 116
Date solved : Friday, March 14, 2025 at 12:55:54 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{4} y^{\prime \prime }&=\left (y-x y^{\prime }\right )^{3} \end{align*}

✓ Maple. Time used: 0.056 (sec). Leaf size: 37

ode:=x^4*diff(diff(y(x),x),x) = (y(x)-x*diff(y(x),x))^3; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 60.191 (sec). Leaf size: 95

ode=x^4*D[y[x],{x,2}]==(y[x]-x*D[y[x],x])^3; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -i x \log \left (\frac {e^{c_2}-\sqrt {e^{2 c_2}-8 i c_1 x^2}}{4 c_1 x}\right ) \\ y(x)\to -i x \log \left (\frac {\sqrt {e^{2 c_2}-8 i c_1 x^2}+e^{c_2}}{4 c_1 x}\right ) \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**4*Derivative(y(x), (x, 2)) - (-x*Derivative(y(x), x) + y(x))**3,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -(-(x**4*Derivative(y(x), (x, 2)) - y(x)**3)/x**3 - y(x)**3/x**3)**(1/3) + Derivative(y(x), x) - y(x)/x cannot be solved by the factorable group method