64.14.17 problem 17

Internal problem ID [13576]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 6, Series solutions of linear differential equations. Section 6.1. Exercises page 232
Problem number : 17
Date solved : Tuesday, January 28, 2025 at 05:53:13 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x y^{\prime \prime }+y^{\prime }+2 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 1 \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=2\\ y^{\prime }\left (1\right )&=4 \end{align*}

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 20

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

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

✓ Solution by Mathematica

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

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

\[ y(x)\to \frac {2}{15} (x-1)^5-\frac {1}{3} (x-1)^4+\frac {4}{3} (x-1)^3-4 (x-1)^2+4 (x-1)+2 \]