12.15.55 problem 51

Internal problem ID [2053]
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 II. Exercises 7.6. Page 374
Problem number : 51
Date solved : Tuesday, March 04, 2025 at 01:48:56 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} x \left (x^{2}+1\right ) y^{\prime \prime }+\left (-x^{2}+1\right ) y^{\prime }-8 x y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 32

Order:=6; 
ode:=x*(x^2+1)*diff(diff(y(x),x),x)+(-x^2+1)*diff(y(x),x)-8*x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

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

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

ode=x*(1+x^2)*D[y[x],{x,2}]+(1-x^2)*D[y[x],x]-8*x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (x^4+2 x^2+1\right )+c_2 \left (-\frac {3 x^4}{2}-\frac {3 x^2}{2}+\left (x^4+2 x^2+1\right ) \log (x)\right ) \]

✓ Sympy. Time used: 1.055 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(x**2 + 1)*Derivative(y(x), (x, 2)) - 8*x*y(x) + (1 - x**2)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} \left (\frac {512 x^{5}}{225} + \frac {64 x^{4}}{9} + \frac {128 x^{3}}{9} + 16 x^{2} + 8 x + 1\right ) + O\left (x^{6}\right ) \]