56.4.17 problem 17

Internal problem ID [8906]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 17
Date solved : Monday, January 27, 2025 at 05:19:03 PM
CAS classification : [_Lienard]

\begin{align*} x y^{\prime \prime }+2 y^{\prime }+y x&=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: 0.007 (sec). Leaf size: 14

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

\[ y = 1-\frac {1}{6} x^{2}+\frac {1}{120} x^{4}+\operatorname {O}\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.204 (sec). Leaf size: 19

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

\[ y(x)\to \frac {x^4}{120}-\frac {x^2}{6}+1 \]