23.5.63 problem 63

Internal problem ID [6672]
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 : 63
Date solved : Friday, October 03, 2025 at 02:09:38 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 104

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

\[ y = -\frac {3 \left (\left (-3 \Gamma \left (\frac {2}{3}\right ) {\mathrm e}^{\frac {x^{3}}{3}} c_3 -\frac {x \left (c_2 x +c_1 \right )}{3}\right ) \left (-x^{3}\right )^{{2}/{3}}+c_3 \,x^{3} \left (\Gamma \left (\frac {1}{3}, -\frac {x^{3}}{3}\right ) \Gamma \left (\frac {2}{3}\right ) \left (-x^{3}\right )^{{1}/{3}} 3^{{1}/{3}}-\frac {2 \,3^{{5}/{6}} \pi \left (-x^{3}\right )^{{1}/{3}}}{3}-2 \Gamma \left (\frac {2}{3}\right ) \Gamma \left (\frac {2}{3}, -\frac {x^{3}}{3}\right ) 3^{{2}/{3}}+2 \,3^{{2}/{3}} \Gamma \left (\frac {2}{3}\right )^{2}\right )\right )}{\left (-x^{3}\right )^{{2}/{3}}} \]

✓ Mathematica. Time used: 0.042 (sec). Leaf size: 79

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

\begin{align*} y(x)&\to \frac {1}{18} \left (-\sqrt [3]{3} c_3 \left (-x^3\right )^{2/3} \Gamma \left (-\frac {2}{3},-\frac {x^3}{3}\right )+3^{2/3} c_3 \sqrt [3]{-x^3} \Gamma \left (-\frac {1}{3},-\frac {x^3}{3}\right )+9 x (c_2 x+2 c_1)\right ) \end{align*}

✗ 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) - 2*y(x) + Derivative(y(x), (x, 3)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (x**2*Derivative(y(x), (x, 2)) + 2*y(x) -