62.30.1 problem Ex 1

Internal problem ID [12894]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter VIII, Linear differential equations of the second order. Article 53. Change of dependent variable. Page 125
Problem number : Ex 1
Date solved : Monday, March 31, 2025 at 07:23:49 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.006 (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: 0.217 (sec). Leaf size: 114

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]
 

\[ y(x)\to x \int _1^x\frac {e^{-\frac {1}{3} K[1]^3} \Gamma \left (-\frac {1}{3},-\frac {1}{3} K[1]^3\right ) K[1] \sqrt [3]{-K[1]^3}}{3 \sqrt [3]{3}}dK[1]-\frac {e^{-\frac {x^3}{3}} \sqrt [3]{-x^3} \left (-1+c_2 e^{\frac {x^3}{3}}\right ) \Gamma \left (-\frac {1}{3},-\frac {x^3}{3}\right )}{3 \sqrt [3]{3}}+c_1 x \]

✗ 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