6.16.41 problem 37

Internal problem ID [2103]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF FROBENIUS III. Exercises 7.7. Page 389
Problem number : 37
Date solved : Tuesday, September 30, 2025 at 05:24:03 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (-x^{2}+5\right ) y^{\prime }-\left (25 x^{2}+7\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 35

Order:=6; 
ode:=x^2*(x^2+1)*diff(diff(y(x),x),x)-x*(-x^2+5)*diff(y(x),x)-(25*x^2+7)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{7} \left (1-\frac {6}{5} x^{2}+\frac {7}{5} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (-203212800+406425600 x^{2}-609638400 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 40

ode=x^2*(1+x^2)*D[y[x],{x,2}]-x*(5-x^2)*D[y[x],x]-(7+25*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (3 x^3-2 x+\frac {1}{x}\right )+c_2 \left (\frac {7 x^{11}}{5}-\frac {6 x^9}{5}+x^7\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(x**2 + 1)*Derivative(y(x), (x, 2)) - x*(5 - x**2)*Derivative(y(x), x) - (25*x**2 + 7)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : Expected Expr or iterable but got None