54.9.16 problem 16

Internal problem ID [8686]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. Miscellaneous Exercises. page 394
Problem number : 16
Date solved : Wednesday, March 05, 2025 at 06:12:25 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 37

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

\[ y = \left (x^{2}+1\right ) y \left (0\right )+\left (x +\frac {1}{3} x^{3}-\frac {1}{15} x^{5}+\frac {1}{35} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 38

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

\[ y(x)\to c_1 \left (x^2+1\right )+c_2 \left (\frac {x^7}{35}-\frac {x^5}{15}+\frac {x^3}{3}+x\right ) \]

✓ Sympy. Time used: 0.774 (sec). Leaf size: 27

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

\[ y{\left (x \right )} = C_{2} \left (x^{2} + 1\right ) + C_{1} x \left (- \frac {x^{4}}{15} + \frac {x^{2}}{3} + 1\right ) + O\left (x^{8}\right ) \]