83.34.5 problem 5

Internal problem ID [19347]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VII. Exact differential equations and certain particular forms of equations. Exercise VII (H) at page 118
Problem number : 5
Date solved : Monday, March 31, 2025 at 07:09:10 PM
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 }&=\left (x^{3}+2 x y\right ) y^{\prime }-4 y^{2} \end{align*}

✓ Maple. Time used: 0.028 (sec). Leaf size: 22

ode:=x^4*diff(diff(y(x),x),x) = (x^3+2*x*y(x))*diff(y(x),x)-4*y(x)^2; 
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: 70.828 (sec). Leaf size: 83

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

\[ y(x)\to \frac {x^2 \left (\left (1-i \sqrt {-1-c_1}\right ) x^{2 i \sqrt {-1-c_1}}+\left (1+i \sqrt {-1-c_1}\right ) c_2\right )}{x^{2 i \sqrt {-1-c_1}}+c_2} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**4*Derivative(y(x), (x, 2)) - (x**3 + 2*x*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