58.2.27 problem 27
Internal
problem
ID
[9150]
Book
:
Second
order
enumerated
odes
Section
:
section
2
Problem
number
:
27
Date
solved
:
Wednesday, March 05, 2025 at 07:34:57 AM
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: 0.026 (sec). Leaf size: 192
ode:=diff(diff(y(x),x),x)-x^2*diff(y(x),x)+x*y(x) = x^(m+1);
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\left (\operatorname {WhittakerM}\left (\frac {m}{6}, \frac {m}{6}+\frac {1}{2}, \frac {x^{3}}{3}\right ) {\mathrm e}^{-\frac {x^{3}}{6}} 3^{\frac {5}{3}+\frac {m}{6}} \left (x^{3}\right )^{-\frac {m}{6}} x^{m}-3 \,3^{{2}/{3}} c_{1} \left (m +3\right )\right ) \left (\Gamma \left (\frac {2}{3}, -\frac {x^{3}}{3}\right )-\Gamma \left (\frac {2}{3}\right )\right ) \left (-x^{3}\right )^{{1}/{3}}-9 \operatorname {WhittakerM}\left (\frac {m}{6}, \frac {m}{6}+\frac {1}{2}, \frac {x^{3}}{3}\right ) \left (x^{3}\right )^{-\frac {m}{6}} 3^{\frac {m}{6}} x^{m} {\mathrm e}^{\frac {x^{3}}{6}}+\left (\left (\int x^{m +1} \left (\left (-x^{3}\right )^{{1}/{3}} 3^{{2}/{3}} \Gamma \left (\frac {2}{3}\right ) {\mathrm e}^{-\frac {x^{3}}{3}}-\left (-x^{3}\right )^{{1}/{3}} 3^{{2}/{3}} \Gamma \left (\frac {2}{3}, -\frac {x^{3}}{3}\right ) {\mathrm e}^{-\frac {x^{3}}{3}}+3\right )d x +3 c_{2} \right ) x +9 c_{1} {\mathrm e}^{\frac {x^{3}}{3}}\right ) \left (m +3\right )}{3 m +9}
\]
✓ Mathematica. Time used: 0.277 (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]
\[
y(x)\to x \int _1^x\frac {e^{-\frac {1}{3} K[1]^3} \Gamma \left (-\frac {1}{3},-\frac {1}{3} K[1]^3\right ) K[1]^{m+1} \sqrt [3]{-K[1]^3}}{3 \sqrt [3]{3}}dK[1]-\frac {\sqrt [3]{-x^3} \left (x^3\right )^{-m/3} \Gamma \left (-\frac {1}{3},-\frac {x^3}{3}\right ) \left (-3^{m/3} x^m \Gamma \left (\frac {m+3}{3},\frac {x^3}{3}\right )+c_2 \left (x^3\right )^{m/3}\right )}{3 \sqrt [3]{3}}+c_1 x
\]
✗ 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