68.1.11 problem Problem 1.8(b)

Internal problem ID [14069]
Book : Differential Equations, Linear, Nonlinear, Ordinary, Partial. A.C. King, J.Billingham, S.R.Otto. Cambridge Univ. Press 2003
Section : Chapter 1 VARIABLE COEFFICIENT, SECOND ORDER DIFFERENTIAL EQUATIONS. Problems page 28
Problem number : Problem 1.8(b)
Date solved : Monday, March 31, 2025 at 12:02:34 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+4 y^{\prime }-x y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 32

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

\[ y = c_1 \left (1+\frac {1}{10} x^{2}+\frac {1}{280} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (12-6 x^{2}-\frac {3}{2} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{3}} \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 42

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

\[ y(x)\to c_1 \left (\frac {1}{x^3}-\frac {x}{8}-\frac {1}{2 x}\right )+c_2 \left (\frac {x^4}{280}+\frac {x^2}{10}+1\right ) \]

✓ Sympy. Time used: 0.904 (sec). Leaf size: 46

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

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4}}{280} + \frac {x^{2}}{10} + 1\right ) + \frac {C_{1} \left (- \frac {x^{8}}{5760} - \frac {x^{6}}{144} - \frac {x^{4}}{8} - \frac {x^{2}}{2} + 1\right )}{x^{3}} + O\left (x^{6}\right ) \]