12.14.42 problem 44

Internal problem ID [1983]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.5 THE METHOD OF FROBENIUS I. Exercises 7.5. Page 358
Problem number : 44
Date solved : Tuesday, March 04, 2025 at 01:47:29 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x^{2}+2\right ) y^{\prime \prime }+x \left (x^{2}+3\right ) y^{\prime }-y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 36

Order:=6; 
ode:=x^2*(x^2+2)*diff(diff(y(x),x),x)+x*(x^2+3)*diff(y(x),x)-y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_2 \,x^{{3}/{2}} \left (1-\frac {1}{56} x^{2}+\frac {25}{9856} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \left (1-\frac {1}{2} x^{2}+\frac {1}{40} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 50

ode=x^2*(2+x^2)*D[y[x],{x,2}]+x*(3+x^2)*D[y[x],x]-y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt {x} \left (\frac {25 x^4}{9856}-\frac {x^2}{56}+1\right )+\frac {c_2 \left (\frac {x^4}{40}-\frac {x^2}{2}+1\right )}{x} \]

✓ Sympy. Time used: 1.022 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(x**2 + 2)*Derivative(y(x), (x, 2)) + x*(x**2 + 3)*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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