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 : Monday, January 27, 2025 at 08:06:02 AM
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*}

✓ Solution by Maple

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

Order:=6; 
dsolve([4*diff(y(x),x$2)+x*diff(y(x),x)+4*y(x)=0,y(0) = 1, D(y)(0) = 0],y(x),type='series',x=0);
 

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

✓ Solution by Mathematica

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

AsymptoticDSolveValue[{4*D[y[x],{x,2}]+x*D[y[x],x]+4*y[x]==0,{y[0]==1,Derivative[1][y][0] ==0}},y[x],{x,0,"6"-1}]
 

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