67.24.20 problem 34.7 (f)

Internal problem ID [16987]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 34. Power series solutions II: Generalization and theory. Additional Exercises. page 678
Problem number : 34.7 (f)
Date solved : Thursday, October 02, 2025 at 01:41:32 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 59

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

\[ y = \left (1-x^{2}-\frac {5}{3} x^{3}+\frac {11}{12} x^{4}+\frac {101}{60} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}-\frac {1}{2} x^{3}-\frac {9}{8} x^{4}+\frac {41}{40} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 68

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

\[ y(x)\to c_1 \left (\frac {101 x^5}{60}+\frac {11 x^4}{12}-\frac {5 x^3}{3}-x^2+1\right )+c_2 \left (\frac {41 x^5}{40}-\frac {9 x^4}{8}-\frac {x^3}{2}-\frac {x^2}{2}+x\right ) \]

✓ Sympy. Time used: 0.302 (sec). Leaf size: 46

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

\[ y{\left (x \right )} = C_{2} \left (\frac {11 x^{4}}{12} - \frac {5 x^{3}}{3} - x^{2} + 1\right ) + C_{1} x \left (- \frac {9 x^{3}}{8} - \frac {x^{2}}{2} - \frac {x}{2} + 1\right ) + O\left (x^{6}\right ) \]