20.24.18 problem Problem 19

Internal problem ID [4003]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.2. page 739
Problem number : Problem 19
Date solved : Tuesday, March 04, 2025 at 05:22:30 PM
CAS classification : [_Lienard]

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

Using series method with expansion around

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

With initial conditions

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 14

Order:=6; 
ode:=4*diff(diff(y(x),x),x)+x*diff(y(x),x)+4*y(x) = 0; 
ic:=y(0) = 1, D(y)(0) = 0; 
dsolve([ode,ic],y(x),type='series',x=0);
 

\[ y \left (x \right ) = 1-\frac {1}{2} x^{2}+\frac {1}{16} x^{4}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 19

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

\[ y(x)\to \frac {x^4}{16}-\frac {x^2}{2}+1 \]

✓ Sympy. Time used: 0.779 (sec). Leaf size: 31

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

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4}}{16} - \frac {x^{2}}{2} + 1\right ) + C_{1} x \left (1 - \frac {5 x^{2}}{24}\right ) + O\left (x^{6}\right ) \]