81.6.13 problem 13

Internal problem ID [18625]
Book : A short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907
Section : Chapter V. Homogeneous linear differential equations. Exact equations. Exercises at page 69
Problem number : 13
Date solved : Monday, March 31, 2025 at 05:46:56 PM
CAS classification : [[_3rd_order, _fully, _exact, _linear]]

\begin{align*} \left (x^{3}+x^{2}-3 x +1\right ) y^{\prime \prime \prime }+\left (9 x^{2}+6 x -9\right ) y^{\prime \prime }+\left (18 x +6\right ) y^{\prime }+6 y&=x^{3} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 38

ode:=(x^3+x^2-3*x+1)*diff(diff(diff(y(x),x),x),x)+(9*x^2+6*x-9)*diff(diff(y(x),x),x)+(18*x+6)*diff(y(x),x)+6*y(x) = x^3; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {x^{6}+120 c_1 \,x^{2}+120 c_2 x +120 c_3}{120 \left (x -1\right ) \left (x^{2}+2 x -1\right )} \]

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 42

ode=(x^3+x^2-3*x+1)*D[y[x],{x,3}]+(9*x^2+6*x-9)*D[y[x],{x,2}]+(18*x+6)*D[y[x],x]+6*y[x]==x^3; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {x^6+60 c_3 x^2+120 c_2 x+120 c_1}{120 \left (x^3+x^2-3 x+1\right )} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**3 + (18*x + 6)*Derivative(y(x), x) + (9*x**2 + 6*x - 9)*Derivative(y(x), (x, 2)) + (x**3 + x**2 - 3*x + 1)*Derivative(y(x), (x, 3)) + 6*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-x**3*Derivative(y(x), (x, 3)) + x**3 - 9*x**2*Derivative(y(x), (x, 2)) - x**2*Derivative(y(x), (x, 3)) - 6*x*Derivative(y(x), (x, 2)) + 3*x*Derivative(y(x), (x, 3)) - 6*y(x) + 9*Derivative(y(x), (x, 2)) - Derivative(y(x), (x, 3)))/(6*(3*x + 1)) cannot be solved by the factorable group method