75.2.6 problem 6

Internal problem ID [19900]
Book : A short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907
Section : Chapter II. Change of variable. Exercises at page 20
Problem number : 6
Date solved : Friday, October 03, 2025 at 07:34:18 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} y^{2} y^{\prime \prime \prime }-\left (3 y y^{\prime }+2 x y^{2}\right ) y^{\prime \prime }+\left (2 {y^{\prime }}^{2}+2 x y y^{\prime }+3 x^{2} y^{2}\right ) y^{\prime }+x^{3} y^{3}&=0 \end{align*}

✓ Maple. Time used: 0.360 (sec). Leaf size: 557

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

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Mathematica

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

Not solved

✗ Sympy

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

NotImplementedError : The given ODE x*y(x)/3 + (-7*x**2*y(x)**2/2 + 9*y(x)*Derivative(y(x), (x, 2))/2)/(3*(-1/2 + sqrt(3)*I/2)*(31*x**3*y(x)**3/4 - 9*x*(3*x**2*y(x)**2/2 - 3*y(x)*Derivative(y(x), (x, 2))/2)*y(x)/2 - 27*x*y(x)**2*Derivative(y(x), (x, 2))/2 + sqrt(-4*(-7*x**2*y(x)**2/2 + 9*y(x)*Derivative(y(x), (x, 2))/2)**3 + (31*x**3*y(x)**3/2 - 9*x*(3*x**2*y(x)**2/2 - 3*y(x)*Derivative(y(x), (x, 2))/2)*y(x) - 27*x*y(x)**2*Derivative(y(x), (x, 2)) + 27*y(x)**2*Derivative(y(x), (x, 3))/2)**2)/2 + 27*y(x)**2*Derivative(y(x), (x, 3))/4)**(1/3)) + (-1/2 + sqrt(3)*I/2)*(31*x**3*y(x)**3/4 - 9*x*(3*x**2*y(x)**2/2 - 3*y(x)*Derivative(y(x), (x, 2))/2)*y(x)/2 - 27*x*y(x)**2*Derivative(y(x), (x, 2))/2 + sqrt(-4*(-7*x**2*y(x)**2/2 + 9*y(x)*Derivative(y(x), (x, 2))/2)**3 + (31*x**3*y(x)**3/2 - 9*x*(3*x**2*y(x)**2/2 - 3*y(x)*Derivative(y(x), (x, 2))/2)*y(x) - 27*x*y(x)**2*Derivative(y(x), (x, 2)) + 27*y(x)**2*Derivative(y(x), (x, 3))/2)**2)/2 + 27*y(x)**2*Derivative(y(x), (x, 3))/4)**(1/3)/3 + Derivative(y(x), x) cannot be solved by the factorable group method