64.15.15 problem 15

Internal problem ID [13521]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 6, Series solutions of linear differential equations. Section 6.2 (Frobenius). Exercises page 251
Problem number : 15
Date solved : Monday, March 31, 2025 at 08:00:15 AM
CAS classification : [_Lienard]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.022 (sec). Leaf size: 32

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

\[ y = c_1 \,x^{3} \left (1+\frac {1}{5} x^{2}+\frac {1}{35} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (12+6 x^{2}+\frac {3}{2} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 44

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

\[ y(x)\to c_1 \left (\frac {x^4}{8}+\frac {x^2}{2}+1\right )+c_2 \left (\frac {x^7}{35}+\frac {x^5}{5}+x^3\right ) \]

✓ Sympy. Time used: 0.868 (sec). Leaf size: 31

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

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