82.49.2 problem Ex. 2

Internal problem ID [18933]
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 page 111
Problem number : Ex. 2
Date solved : Monday, March 31, 2025 at 06:25:25 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 54

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

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

✓ Mathematica. Time used: 0.246 (sec). Leaf size: 42

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

\[ y(x)\to -\frac {c_2 \sqrt [3]{-x^3} \Gamma \left (-\frac {1}{3},-\frac {x^3}{3}\right )}{3 \sqrt [3]{3}}+c_1 x+1 \]

✗ Sympy

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

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