15.24.15 problem 15

Internal problem ID [3387]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 42, page 206
Problem number : 15
Date solved : Sunday, March 30, 2025 at 01:38:46 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 46

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

\[ y = \frac {c_1 \,x^{4} \left (1+\frac {1}{12} x^{2}+\frac {1}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (27 x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144-108 x^{2}-36 x^{4}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{3}} \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 53

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

\[ y(x)\to c_2 \left (\frac {x^5}{128}+\frac {x^3}{12}+x\right )+c_1 \left (\frac {19 x^4+48 x^2+64}{64 x^3}-\frac {3}{16} x \log (x)\right ) \]

✓ Sympy. Time used: 0.918 (sec). Leaf size: 20

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

\[ y{\left (x \right )} = C_{1} x \left (\frac {x^{4}}{128} + \frac {x^{2}}{12} + 1\right ) + O\left (x^{6}\right ) \]