6.14.33 problem 35

Internal problem ID [1974]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.5 THE METHOD OF FROBENIUS I. Exercises 7.5. Page 358
Problem number : 35
Date solved : Tuesday, September 30, 2025 at 05:22:00 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-2 x \left (-x^{2}+2\right ) y^{\prime }+4 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 33

Order:=6; 
ode:=x^2*(x^2+1)*diff(diff(y(x),x),x)-2*x*(-x^2+2)*diff(y(x),x)+4*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{4} \left (1-2 x^{2}+3 x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 x \left (12+12 x^{2}-36 x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 34

ode=x^2*(1+x^2)*D[y[x],{x,2}]-2*x*(2-x^2)*D[y[x],x]+4*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (-3 x^5+x^3+x\right )+c_2 \left (3 x^8-2 x^6+x^4\right ) \]

✓ Sympy. Time used: 0.445 (sec). Leaf size: 14

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(x**2 + 1)*Derivative(y(x), (x, 2)) - 2*x*(2 - x**2)*Derivative(y(x), x) + 4*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x^{4} + C_{1} x + O\left (x^{6}\right ) \]