12.13.31 problem 31(e)

Internal problem ID [1922]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.3 SERIES SOLUTIONS NEAR AN ORDINARY POINT II. Exercises 7.3. Page 338
Problem number : 31(e)
Date solved : Tuesday, March 04, 2025 at 01:46:16 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (3 x^{2}+8 x +4\right ) y^{\prime \prime }+\left (16+12 x \right ) y^{\prime }+6 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 59

Order:=6; 
ode:=(3*x^2+8*x+4)*diff(diff(y(x),x),x)+(16+12*x)*diff(y(x),x)+6*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {3}{4} x^{2}+\frac {3}{2} x^{3}-\frac {39}{16} x^{4}+\frac {15}{4} x^{5}\right ) y \left (0\right )+\left (x -2 x^{2}+\frac {13}{4} x^{3}-5 x^{4}+\frac {121}{16} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 66

ode=(4+8*x+3*x^2)*D[y[x],{x,2}]+(16+12*x)*D[y[x],x]+6*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {15 x^5}{4}-\frac {39 x^4}{16}+\frac {3 x^3}{2}-\frac {3 x^2}{4}+1\right )+c_2 \left (\frac {121 x^5}{16}-5 x^4+\frac {13 x^3}{4}-2 x^2+x\right ) \]

✓ Sympy. Time used: 0.803 (sec). Leaf size: 49

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((12*x + 16)*Derivative(y(x), x) + (3*x**2 + 8*x + 4)*Derivative(y(x), (x, 2)) + 6*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {39 x^{4}}{16} + \frac {3 x^{3}}{2} - \frac {3 x^{2}}{4} + 1\right ) + C_{1} x \left (- 5 x^{3} + \frac {13 x^{2}}{4} - 2 x + 1\right ) + O\left (x^{6}\right ) \]