23.5.165 problem 165

Internal problem ID [6774]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 5. THE EQUATION IS LINEAR AND OF ORDER GREATER THAN TWO, page 410
Problem number : 165
Date solved : Tuesday, September 30, 2025 at 03:51:34 PM
CAS classification : [[_high_order, _missing_y]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 51

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

\[ y = -\left (x^{2}+2 x +\frac {2}{3}\right ) c_1 \,\operatorname {Ei}_{1}\left (\frac {2}{x}\right )+\frac {x c_1 \left (x^{2}+5 x +2\right ) {\mathrm e}^{-\frac {2}{x}}}{6}+\frac {c_2 \,x^{2}}{2}+c_3 x +c_4 \]

✓ Mathematica. Time used: 0.109 (sec). Leaf size: 60

ode=x^2*D[y[x],{x,4}] == 2*D[y[x],{x,3}]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{6} c_1 \left (2 \left (3 x^2+6 x+2\right ) \operatorname {ExpIntegralEi}\left (-\frac {2}{x}\right )+e^{-2/x} x \left (x^2+5 x+2\right )\right )+c_4 x^2+c_3 x+c_2 \end{align*}

✓ Sympy. Time used: 1.696 (sec). Leaf size: 83

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 4)) - 2*Derivative(y(x), (x, 3)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} + \frac {C_{3} x^{3} e^{- \frac {2}{x}}}{6} + \frac {2 C_{3} \operatorname {Ei}{\left (\frac {2 e^{i \pi }}{x} \right )}}{3} + x^{2} \left (C_{3} \operatorname {Ei}{\left (\frac {2 e^{i \pi }}{x} \right )} + \frac {5 C_{3} e^{- \frac {2}{x}}}{6} + C_{4}\right ) + x \left (C_{2} + 2 C_{3} \operatorname {Ei}{\left (\frac {2 e^{i \pi }}{x} \right )} + \frac {C_{3} e^{- \frac {2}{x}}}{3}\right ) \]