14.25.10 problem 11

Internal problem ID [4015]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.4. page 758
Problem number : 11
Date solved : Tuesday, September 30, 2025 at 07:00:25 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 6 x^{2} y^{\prime \prime }+x \left (1+18 x \right ) y^{\prime }+\left (1+12 x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 47

Order:=6; 
ode:=6*x^2*diff(diff(y(x),x),x)+x*(1+18*x)*diff(y(x),x)+(1+12*x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{{1}/{3}} \left (1-\frac {18}{5} x +\frac {324}{55} x^{2}-\frac {5832}{935} x^{3}+\frac {104976}{21505} x^{4}-\frac {1889568}{623645} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \sqrt {x}\, \left (1-3 x +\frac {9}{2} x^{2}-\frac {9}{2} x^{3}+\frac {27}{8} x^{4}-\frac {81}{40} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 88

ode=6*x^2*D[y[x],{x,2}]+x*(1+18*x)*D[y[x],x]+(1+12*x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt {x} \left (-\frac {81 x^5}{40}+\frac {27 x^4}{8}-\frac {9 x^3}{2}+\frac {9 x^2}{2}-3 x+1\right )+c_2 \sqrt [3]{x} \left (-\frac {1889568 x^5}{623645}+\frac {104976 x^4}{21505}-\frac {5832 x^3}{935}+\frac {324 x^2}{55}-\frac {18 x}{5}+1\right ) \]

✓ Sympy. Time used: 0.400 (sec). Leaf size: 71

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(6*x**2*Derivative(y(x), (x, 2)) + x*(18*x + 1)*Derivative(y(x), x) + (12*x + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \sqrt {x} \left (\frac {27 x^{4}}{8} - \frac {9 x^{3}}{2} + \frac {9 x^{2}}{2} - 3 x + 1\right ) + C_{1} \sqrt [3]{x} \left (\frac {104976 x^{4}}{21505} - \frac {5832 x^{3}}{935} + \frac {324 x^{2}}{55} - \frac {18 x}{5} + 1\right ) + O\left (x^{6}\right ) \]