62.35.1 problem Ex 1

Internal problem ID [12918]
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 59. Linear equations with particular integral known. Page 136
Problem number : Ex 1
Date solved : Friday, March 14, 2025 at 12:17:17 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.027 (sec). Leaf size: 17

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

\[ y = c_{1} x +c_{2} x^{2}+c_{3} {\mathrm e}^{x} \]

✓ Mathematica. Time used: 0.079 (sec). Leaf size: 80

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

\[ y(x)\to \frac {1}{2} x \left (2 c_3 \int _1^x\frac {1}{2} \int \exp \left (\int _1^{K[2]}\frac {K[1]^3-3 K[1]^2+6 K[1]-6}{K[1] \left (K[1]^2-2 K[1]+2\right )}dK[1]\right ) \, dK[2]dK[2]+c_2 x+2 c_1\right ) \]

✗ Sympy

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

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