10.14.10 problem 7. case \(x_0=0\)

Internal problem ID [1392]
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 : 7. case \(x_0=0\)
Date solved : Tuesday, March 04, 2025 at 12:35:05 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 44

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

\[ y = \left (1-\frac {1}{2} x^{2}+\frac {3}{8} x^{4}+\frac {1}{20} x^{5}\right ) y \left (0\right )+\left (x -\frac {5}{6} x^{3}+\frac {13}{24} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 49

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

\[ y(x)\to c_2 \left (\frac {13 x^5}{24}-\frac {5 x^3}{6}+x\right )+c_1 \left (\frac {x^5}{20}+\frac {3 x^4}{8}-\frac {x^2}{2}+1\right ) \]

✓ Sympy. Time used: 1.762 (sec). Leaf size: 32

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

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