1.15.3 problem 3

Internal problem ID [459]
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 : 3
Date solved : Tuesday, September 30, 2025 at 03:59:00 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 83

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

\[ y(x)\to c_2 e^{\frac {1}{x}} \left (\frac {58893191 x^5}{57600}+\frac {194987 x^4}{1152}+\frac {4813 x^3}{144}+\frac {65 x^2}{8}+\frac {5 x}{2}+1\right ) x^2+c_1 \left (\frac {37 x^5}{30}-\frac {x^4}{2}+\frac {x^3}{3}-\frac {x^2}{2}+1\right ) \]

✗ Sympy

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