12.14.35 problem 37

Internal problem ID [1976]
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 : 37
Date solved : Tuesday, March 04, 2025 at 01:47:20 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }+x \left (-19 x^{2}+7\right ) y^{\prime }-\left (14 x^{2}+1\right ) 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:=4*x^2*(-x^2+1)*diff(diff(y(x),x),x)+x*(-19*x^2+7)*diff(y(x),x)-(14*x^2+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_2 \,x^{{5}/{4}} \left (1+\frac {9}{13} x^{2}+\frac {51}{91} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \left (1+\frac {1}{2} x^{2}+\frac {3}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

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

ode=4*x^2*(1-x^2)*D[y[x],{x,2}]+x*(7-19*x^2)*D[y[x],x]-(1+14*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt [4]{x} \left (\frac {51 x^4}{91}+\frac {9 x^2}{13}+1\right )+\frac {c_2 \left (\frac {3 x^4}{8}+\frac {x^2}{2}+1\right )}{x} \]

✓ Sympy. Time used: 1.172 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**2*(1 - x**2)*Derivative(y(x), (x, 2)) + x*(7 - 19*x**2)*Derivative(y(x), x) - (14*x**2 + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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