14.25.16 problem 17

Internal problem ID [4021]
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 : 17
Date solved : Tuesday, September 30, 2025 at 07:00:30 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} 3 x^{2} y^{\prime \prime }+x \left (7+3 x \right ) y^{\prime }+\left (1+6 x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 47

Order:=6; 
ode:=3*x^2*diff(diff(y(x),x),x)+x*(7+3*x)*diff(y(x),x)+(1+6*x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \left (1-3 x +\frac {9}{4} x^{2}-\frac {27}{28} x^{3}+\frac {81}{280} x^{4}-\frac {243}{3640} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x}+\frac {c_2 \left (1-x +\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{24} x^{4}-\frac {1}{120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{1}/{3}}} \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 84

ode=3*x^2*D[y[x],{x,2}]+x*(7+3*x)*D[y[x],x]+(1+6*x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {c_1 \left (-\frac {x^5}{120}+\frac {x^4}{24}-\frac {x^3}{6}+\frac {x^2}{2}-x+1\right )}{\sqrt [3]{x}}+\frac {c_2 \left (-\frac {243 x^5}{3640}+\frac {81 x^4}{280}-\frac {27 x^3}{28}+\frac {9 x^2}{4}-3 x+1\right )}{x} \]

✓ Sympy. Time used: 0.402 (sec). Leaf size: 78

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*x**2*Derivative(y(x), (x, 2)) + x*(3*x + 7)*Derivative(y(x), x) + (6*x + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = \frac {C_{2} \left (- \frac {x^{5}}{120} + \frac {x^{4}}{24} - \frac {x^{3}}{6} + \frac {x^{2}}{2} - x + 1\right )}{\sqrt [3]{x}} + \frac {C_{1} \left (\frac {729 x^{6}}{58240} - \frac {243 x^{5}}{3640} + \frac {81 x^{4}}{280} - \frac {27 x^{3}}{28} + \frac {9 x^{2}}{4} - 3 x + 1\right )}{x} + O\left (x^{6}\right ) \]