56.4.42 problem 39
Internal
problem
ID
[8931]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
4.0
Problem
number
:
39
Date
solved
:
Sunday, March 30, 2025 at 01:55:38 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} 2 x^{2} y^{\prime \prime }+x y^{\prime }+\left (x -5\right ) y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Maple. Time used: 0.024 (sec). Leaf size: 277
Order:=6;
ode:=2*x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)+(x-5)*y(x) = 0;
dsolve(ode,y(x),type='series',x=0);
\[
y = x^{{1}/{4}} \left (c_1 \,x^{-\frac {\sqrt {41}}{4}} \left (1+\frac {1}{-2+\sqrt {41}} x +\frac {1}{2} \frac {1}{\left (-2+\sqrt {41}\right ) \left (-4+\sqrt {41}\right )} x^{2}+\frac {1}{6} \frac {1}{\left (-2+\sqrt {41}\right ) \left (-4+\sqrt {41}\right ) \left (-6+\sqrt {41}\right )} x^{3}+\frac {1}{24} \frac {1}{\left (-2+\sqrt {41}\right ) \left (-4+\sqrt {41}\right ) \left (-6+\sqrt {41}\right ) \left (-8+\sqrt {41}\right )} x^{4}+\frac {1}{120} \frac {1}{\left (-2+\sqrt {41}\right ) \left (-4+\sqrt {41}\right ) \left (-6+\sqrt {41}\right ) \left (-8+\sqrt {41}\right ) \left (-10+\sqrt {41}\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \,x^{\frac {\sqrt {41}}{4}} \left (1+\frac {1}{-2-\sqrt {41}} x +\frac {1}{2} \frac {1}{\left (2+\sqrt {41}\right ) \left (4+\sqrt {41}\right )} x^{2}-\frac {1}{6} \frac {1}{\left (2+\sqrt {41}\right ) \left (4+\sqrt {41}\right ) \left (6+\sqrt {41}\right )} x^{3}+\frac {1}{24} \frac {1}{\left (2+\sqrt {41}\right ) \left (4+\sqrt {41}\right ) \left (6+\sqrt {41}\right ) \left (8+\sqrt {41}\right )} x^{4}-\frac {1}{120} \frac {1}{\left (2+\sqrt {41}\right ) \left (4+\sqrt {41}\right ) \left (6+\sqrt {41}\right ) \left (8+\sqrt {41}\right ) \left (10+\sqrt {41}\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )
\]
✓ Mathematica. Time used: 0.007 (sec). Leaf size: 1668
ode=2*x^2*D[y[x],{x,2}]+x*D[y[x],x]+(x-5)*y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
Too large to display
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(2*x**2*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x) + (x - 5)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
NotImplementedError : Not sure of sign of 6 - x2