10.14.5 problem 5. case \(x_0=0\)

Internal problem ID [1387]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 5.3, Series Solutions Near an Ordinary Point, Part II. page 269
Problem number : 5. case \(x_0=0\)
Date solved : Tuesday, March 04, 2025 at 12:34:59 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+6 x y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 52

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

\[ y = \left (1-x^{3}+x^{4}-\frac {4}{5} x^{5}\right ) y \left (0\right )+\left (x -2 x^{2}+\frac {8}{3} x^{3}-\frac {19}{6} x^{4}+\frac {47}{15} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 55

ode=D[y[x],{x,2}]+4*D[y[x],x]+6*x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (-\frac {4 x^5}{5}+x^4-x^3+1\right )+c_2 \left (\frac {47 x^5}{15}-\frac {19 x^4}{6}+\frac {8 x^3}{3}-2 x^2+x\right ) \]

✓ Sympy. Time used: 0.769 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(6*x*y(x) + 4*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)
 

\[ y{\left (x \right )} = C_{2} \left (x^{4} - x^{3} + 1\right ) + C_{1} x \left (- \frac {19 x^{3}}{6} + \frac {8 x^{2}}{3} - 2 x + 1\right ) + O\left (x^{6}\right ) \]