7.15.34 problem 34

Internal problem ID [490]
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 : 34
Date solved : Thursday, March 13, 2025 at 03:36:00 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.085 (sec). Leaf size: 33

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

\[ y = \frac {c_1 \left (1-\frac {7}{24} x^{2}+\frac {19}{3200} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}}+c_2 x \left (1-\frac {1}{42} x^{2}+\frac {1}{1320} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 48

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

\[ y(x)\to c_1 x \left (\frac {x^4}{1320}-\frac {x^2}{42}+1\right )+\frac {c_2 \left (\frac {19 x^4}{3200}-\frac {7 x^2}{24}+1\right )}{\sqrt {x}} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*Derivative(y(x), (x, 2)) - y(x)*cos(x) + 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 2*x**2*Derivative(y(x), (x, 2)) - y(x)*cos(x) + sin(x)*Derivative(y(x), x) does not match hint 2nd_power_series_regular