6.14.36 problem 38

Internal problem ID [1977]
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 : 38
Date solved : Tuesday, September 30, 2025 at 05:22:02 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 3 x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }+x \left (-11 x^{2}+1\right ) y^{\prime }+\left (-5 x^{2}+1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 35

Order:=6; 
ode:=3*x^2*(-x^2+2)*diff(diff(y(x),x),x)+x*(-11*x^2+1)*diff(y(x),x)+(-5*x^2+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{{1}/{3}} \left (1+\frac {4}{11} x^{2}+\frac {40}{253} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \sqrt {x}\, \left (1+\frac {3}{8} x^{2}+\frac {21}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 52

ode=3*x^2*(2-x^2)*D[y[x],{x,2}]+x*(1-11*x^2)*D[y[x],x]+(1-5*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt {x} \left (\frac {21 x^4}{128}+\frac {3 x^2}{8}+1\right )+c_2 \sqrt [3]{x} \left (\frac {40 x^4}{253}+\frac {4 x^2}{11}+1\right ) \]

✓ Sympy. Time used: 0.536 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*x**2*(2 - x**2)*Derivative(y(x), (x, 2)) + x*(1 - 11*x**2)*Derivative(y(x), x) + (1 - 5*x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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