12.14.47 problem 49

Internal problem ID [1988]
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 : 49
Date solved : Tuesday, March 04, 2025 at 01:47:35 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 9 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+3 x \left (13 x^{2}+3\right ) y^{\prime }-\left (-25 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:=9*x^2*(x^2+1)*diff(diff(y(x),x),x)+3*x*(13*x^2+3)*diff(y(x),x)-(-25*x^2+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_2 \,x^{{2}/{3}} \left (1-\frac {3}{4} x^{2}+\frac {9}{14} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \left (1-\frac {2}{3} x^{2}+\frac {5}{9} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{1}/{3}}} \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 52

ode=9*x^2*(1+x^2)*D[y[x],{x,2}]+3*x*(3+13*x^2)*D[y[x],x]-(1-25*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt [3]{x} \left (\frac {9 x^4}{14}-\frac {3 x^2}{4}+1\right )+\frac {c_2 \left (\frac {5 x^4}{9}-\frac {2 x^2}{3}+1\right )}{\sqrt [3]{x}} \]

✓ Sympy. Time used: 1.162 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(9*x**2*(x**2 + 1)*Derivative(y(x), (x, 2)) + 3*x*(13*x**2 + 3)*Derivative(y(x), x) - (1 - 25*x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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