62.36.5 problem Ex 5

Internal problem ID [12932]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IX, Miscellaneous methods for solving equations of higher order than first. Article 60. Exact equation. Integrating factor. Page 139
Problem number : Ex 5
Date solved : Monday, March 31, 2025 at 07:24:59 AM
CAS classification : [[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{3} y y^{\prime \prime \prime }+6 x^{3} y^{\prime } y^{\prime \prime }+18 x^{2} y y^{\prime \prime }+18 x^{2} {y^{\prime }}^{2}+36 x y y^{\prime }+6 y^{2}&=0 \end{align*}

✓ Maple. Time used: 0.025 (sec). Leaf size: 56

ode:=2*x^3*y(x)*diff(diff(diff(y(x),x),x),x)+6*x^3*diff(y(x),x)*diff(diff(y(x),x),x)+18*x^2*y(x)*diff(diff(y(x),x),x)+18*x^2*diff(y(x),x)^2+36*x*y(x)*diff(y(x),x)+6*y(x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ y &= \frac {\sqrt {-x \left (c_1 \,x^{2}+2 c_2 x -2 c_3 \right )}}{x^{2}} \\ y &= -\frac {\sqrt {-x \left (c_1 \,x^{2}+2 c_2 x -2 c_3 \right )}}{x^{2}} \\ \end{align*}

✓ Mathematica. Time used: 0.223 (sec). Leaf size: 60

ode=2*x^3*y[x]*D[y[x],{x,3}]+6*x^3*D[y[x],x]*D[y[x],{x,2}]+18*x^2*y[x]*D[y[x],{x,2}]+18*x^2*D[y[x],x]^2+36*x*y[x]*D[y[x],x]+6*y[x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {\sqrt {c_1 x^2+c_3 x+2 c_2}}{x^{3/2}} \\ y(x)\to \frac {\sqrt {c_1 x^2+c_3 x+2 c_2}}{x^{3/2}} \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**3*y(x)*Derivative(y(x), (x, 3)) + 6*x**3*Derivative(y(x), x)*Derivative(y(x), (x, 2)) + 18*x**2*y(x)*Derivative(y(x), (x, 2)) + 18*x**2*Derivative(y(x), x)**2 + 36*x*y(x)*Derivative(y(x), x) + 6*y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-x**2*Derivative(y(x), (x, 2)) + sqrt(x**4*Derivative(y(x), (x, 2))**2 - 4*x**3*y(x)*Derivative(y(x), (x, 3)) - 24*x**2*y(x)*Derivative(y(x), (x, 2)) + 24*y(x)**2) - 6*y(x))/(6*x) cannot be solved by the factorable group method