83.20.1 problem 4

Internal problem ID [22059]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Examination VII. page 256
Problem number : 4
Date solved : Thursday, October 02, 2025 at 08:23:17 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}+1\right ) y^{\prime \prime }-8 x y^{\prime }+20 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 39

Order:=6; 
ode:=(x^2+1)*diff(diff(y(x),x),x)-8*x*diff(y(x),x)+20*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (5 x^{4}-10 x^{2}+1\right ) y \left (0\right )+\left (x -2 x^{3}+\frac {1}{5} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 36

ode=(1+x^2)*D[y[x],{x,2}]-8*x*D[y[x],x]+20*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {x^5}{5}-2 x^3+x\right )+c_1 \left (5 x^4-10 x^2+1\right ) \]

✓ Sympy. Time used: 0.261 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-8*x*Derivative(y(x), x) + (x**2 + 1)*Derivative(y(x), (x, 2)) + 20*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (5 x^{4} - 10 x^{2} + 1\right ) + C_{1} x \left (1 - 2 x^{2}\right ) + O\left (x^{6}\right ) \]