12.14.11 problem 8

Internal problem ID [1952]
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 : 8
Date solved : Tuesday, March 04, 2025 at 01:46:49 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 18 x^{2} \left (1+x \right ) y^{\prime \prime }+3 x \left (x^{2}+11 x +5\right ) y^{\prime }-\left (-5 x^{2}-2 x +1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 46

Order:=6; 
ode:=18*x^2*(1+x)*diff(diff(y(x),x),x)+3*x*(x^2+11*x+5)*diff(y(x),x)-(-5*x^2-2*x+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_2 \sqrt {x}\, \left (1-\frac {1}{3} x +\frac {2}{15} x^{2}-\frac {5}{63} x^{3}+\frac {23}{405} x^{4}-\frac {458}{10395} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \left (1-\frac {1}{12} x^{2}+\frac {1}{18} x^{3}-\frac {11}{288} x^{4}+\frac {31}{1080} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{1}/{6}}} \]

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

ode=18*x^2*(1+x)*D[y[x],{x,2}]+3*x*(5+11*x+x^2)*D[y[x],x]-(1-2*x-5*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt [3]{x} \left (-\frac {458 x^5}{10395}+\frac {23 x^4}{405}-\frac {5 x^3}{63}+\frac {2 x^2}{15}-\frac {x}{3}+1\right )+\frac {c_2 \left (\frac {31 x^5}{1080}-\frac {11 x^4}{288}+\frac {x^3}{18}-\frac {x^2}{12}+1\right )}{\sqrt [6]{x}} \]

✓ Sympy. Time used: 1.265 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(18*x**2*(x + 1)*Derivative(y(x), (x, 2)) + 3*x*(x**2 + 11*x + 5)*Derivative(y(x), x) - (-5*x**2 - 2*x + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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