6.13.27 problem 31(a)

Internal problem ID [1918]
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(a)
Date solved : Tuesday, September 30, 2025 at 05:21:17 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (2 x^{2}+3 x +1\right ) y^{\prime \prime }+\left (6+8 x \right ) y^{\prime }+4 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 59

Order:=6; 
ode:=(2*x^2+3*x+1)*diff(diff(y(x),x),x)+(6+8*x)*diff(y(x),x)+4*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (30 x^{5}-14 x^{4}+6 x^{3}-2 x^{2}+1\right ) y \left (0\right )+\left (31 x^{5}-15 x^{4}+7 x^{3}-3 x^{2}+x \right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 54

ode=(1+3*x+2*x^2)*D[y[x],{x,2}]+(6+8*x)*D[y[x],x]+4*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (30 x^5-14 x^4+6 x^3-2 x^2+1\right )+c_2 \left (31 x^5-15 x^4+7 x^3-3 x^2+x\right ) \]

✓ Sympy. Time used: 0.273 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((8*x + 6)*Derivative(y(x), x) + (2*x**2 + 3*x + 1)*Derivative(y(x), (x, 2)) + 4*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (- 14 x^{4} + 6 x^{3} - 2 x^{2} + 1\right ) + C_{1} x \left (- 15 x^{3} + 7 x^{2} - 3 x + 1\right ) + O\left (x^{6}\right ) \]