52.2.27 problem 27

Internal problem ID [10450]
Book : Second order enumerated odes
Section : section 2
Problem number : 27
Date solved : Tuesday, September 30, 2025 at 07:27:30 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-x^{2} y^{\prime }+x y&=x^{m +1} \end{align*}

✓ Maple. Time used: 0.027 (sec). Leaf size: 200

ode:=diff(diff(y(x),x),x)-x^2*diff(y(x),x)+x*y(x) = x^(1+m); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (-3 \left (x^{3}\right )^{-\frac {m}{6}} 3^{\frac {m}{6}} {\mathrm e}^{\frac {x^{3}}{6}} \operatorname {WhittakerM}\left (\frac {m}{6}, \frac {m}{6}+\frac {1}{2}, \frac {x^{3}}{3}\right ) x^{m}+\left (3^{{1}/{3}} {\mathrm e}^{\frac {x^{3}}{3}} c_1 +\frac {x \left (\int -\frac {\left (-3 \left (-x^{3}\right )^{{2}/{3}}+x^{3} 3^{{2}/{3}} {\mathrm e}^{-\frac {x^{3}}{3}} \left (\Gamma \left (\frac {2}{3}\right )-\Gamma \left (\frac {2}{3}, -\frac {x^{3}}{3}\right )\right )\right ) x^{m +1}}{\left (-x^{3}\right )^{{2}/{3}}}d x +3 c_2 \right )}{3}\right ) \left (m +3\right )\right ) \left (-x^{3}\right )^{{2}/{3}}-x^{3} \left (\Gamma \left (\frac {2}{3}, -\frac {x^{3}}{3}\right )-\Gamma \left (\frac {2}{3}\right )\right ) \left (3^{\frac {2}{3}+\frac {m}{6}} x^{m} \left (x^{3}\right )^{-\frac {m}{6}} {\mathrm e}^{-\frac {x^{3}}{6}} \operatorname {WhittakerM}\left (\frac {m}{6}, \frac {m}{6}+\frac {1}{2}, \frac {x^{3}}{3}\right )-c_1 \left (m +3\right )\right )}{\left (-x^{3}\right )^{{2}/{3}} \left (m +3\right )} \]

✓ Mathematica. Time used: 0.171 (sec). Leaf size: 144

ode=D[y[x],{x,2}]-x^2*D[y[x],x]+x*y[x]==x^(m+1); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} 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]^{m+1} \sqrt [3]{-K[1]^3}}{3 \sqrt [3]{3}}dK[1]-\frac {\sqrt [3]{-x^3} \left (x^3\right )^{-m/3} \Gamma \left (-\frac {1}{3},-\frac {x^3}{3}\right ) \left (-3^{m/3} x^m \Gamma \left (\frac {m+3}{3},\frac {x^3}{3}\right )+c_2 \left (x^3\right )^{m/3}\right )}{3 \sqrt [3]{3}}+c_1 x \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
m = symbols("m") 
y = Function("y") 
ode = Eq(-x**2*Derivative(y(x), x) + x*y(x) - x**(m + 1) + 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) - x**(m + 1) + Derivative(y(x), (x, 2)))/x**2 cannot be solved by the factorable group method