84.34.3 problem 20.16

Internal problem ID [22330]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 20. Regular singular points and the method of Frobenius. Supplementary problems
Problem number : 20.16
Date solved : Thursday, October 02, 2025 at 08:37:37 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 35

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

\[ y = \frac {c_1 \left (1-\frac {1}{2} x^{2}-\frac {1}{40} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{1}/{3}}}+c_2 \,x^{2} \left (1+\frac {1}{26} x^{2}+\frac {1}{1976} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 50

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

\[ y(x)\to c_1 \left (\frac {x^4}{1976}+\frac {x^2}{26}+1\right ) x^2+\frac {c_2 \left (-\frac {x^4}{40}-\frac {x^2}{2}+1\right )}{\sqrt [3]{x}} \]

✓ Sympy. Time used: 0.322 (sec). Leaf size: 36

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

\[ y{\left (x \right )} = C_{2} x^{2} \left (\frac {x^{2}}{26} + 1\right ) + \frac {C_{1} \left (- \frac {x^{4}}{40} - \frac {x^{2}}{2} + 1\right )}{\sqrt [3]{x}} + O\left (x^{6}\right ) \]