82.54.11 problem Ex. 11

Internal problem ID [18956]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter IX. Equations of the second order. problems at end of chapter at page 120
Problem number : Ex. 11
Date solved : Monday, March 31, 2025 at 06:26:31 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 31

ode:=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);
 

\[ y = \frac {c_3 \left (x +1\right ) {\mathrm e}^{-x}}{2}+x \left (-\frac {c_3 \left (x +2\right ) \operatorname {Ei}_{1}\left (x \right )}{2}+c_2 x +c_1 \right ) \]

✓ Mathematica. Time used: 0.109 (sec). Leaf size: 50

ode=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]
 

\[ y(x)\to \frac {1}{2} \left (-c_3 x (x+2) \operatorname {ExpIntegralEi}(-x)+x ((c_1+2 c_2) x+4 c_2)-c_3 e^{-x} (x+1)\right ) \]

✗ Sympy

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

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