7.15.7 problem 7

Internal problem ID [463]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.3 (Regular singular points). Problems at page 231
Problem number : 7
Date solved : Thursday, March 13, 2025 at 03:35:58 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+6 \sin \left (x \right ) y^{\prime }+6 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.062 (sec). Leaf size: 34

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+6*sin(x)*diff(y(x),x)+6*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \left (1-\frac {1}{3} x^{2}+\frac {1}{200} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) x +c_2 \left (1-\frac {3}{2} x^{2}+\frac {11}{80} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{3}} \]

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 62

ode=x^2*D[y[x],{x,2}]+6*Sin[x]*D[y[x],x]+6*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {c_2 \left (\frac {2 x^6}{11025}+\frac {x^4}{200}-\frac {x^2}{3}+1\right )}{x^2}+\frac {c_1 \left (\frac {11 x^6}{5600}+\frac {11 x^4}{80}-\frac {3 x^2}{2}+1\right )}{x^3} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + 6*y(x) + 6*sin(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : ODE x**2*Derivative(y(x), (x, 2)) + 6*y(x) + 6*sin(x)*Derivative(y(x), x) does not match hint 2nd_power_series_regular