12.14.9 problem 6

Internal problem ID [1950]
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 : 6
Date solved : Tuesday, March 04, 2025 at 01:46:46 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} x^{2} \left (10 x^{2}+x +5\right ) y^{\prime \prime }+x \left (48 x^{2}+3 x +4\right ) y^{\prime }+\left (36 x^{2}+x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 44

Order:=6; 
ode:=x^2*(10*x^2+x+5)*diff(diff(y(x),x),x)+x*(48*x^2+3*x+4)*diff(y(x),x)+(36*x^2+x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{{1}/{5}} \left (1-\frac {6}{25} x -\frac {1217}{625} x^{2}+\frac {41972}{46875} x^{3}+\frac {1447799}{390625} x^{4}-\frac {375253322}{146484375} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1-\frac {1}{4} x -\frac {35}{18} x^{2}+\frac {11}{12} x^{3}+\frac {632}{171} x^{4}-\frac {2671}{1026} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 85

ode=x^2*(5+x+10*x^2)*D[y[x],{x,2}]+x*(4+3*x+48*x^2)*D[y[x],x]+(x+36*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt [5]{x} \left (-\frac {375253322 x^5}{146484375}+\frac {1447799 x^4}{390625}+\frac {41972 x^3}{46875}-\frac {1217 x^2}{625}-\frac {6 x}{25}+1\right )+c_2 \left (-\frac {2671 x^5}{1026}+\frac {632 x^4}{171}+\frac {11 x^3}{12}-\frac {35 x^2}{18}-\frac {x}{4}+1\right ) \]

✓ Sympy. Time used: 1.376 (sec). Leaf size: 14

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(10*x**2 + x + 5)*Derivative(y(x), (x, 2)) + x*(48*x**2 + 3*x + 4)*Derivative(y(x), x) + (36*x**2 + x)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \sqrt [5]{x} + C_{1} + O\left (x^{6}\right ) \]