52.4.8 problem 16

Internal problem ID [8318]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. CHAPTER 6 IN REVIEW. Page 271
Problem number : 16
Date solved : Monday, January 27, 2025 at 03:48:48 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} \left (x +2\right ) y^{\prime \prime }+3 y&=0 \end{align*}

Using series method with expansion around

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 18

Order:=8; 
dsolve([(x+2)*diff(y(x),x$2)+3*y(x)=0,y(0) = 0, D(y)(0) = 1],y(x),type='series',x=0);
 

\[ y = x -\frac {1}{4} x^{3}+\frac {1}{16} x^{4}-\frac {1}{320} x^{6}+\frac {1}{896} x^{7}+\operatorname {O}\left (x^{8}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 33

AsymptoticDSolveValue[{(x+2)*D[y[x],{x,2}]+3*y[x]==0,{y[0]==0,Derivative[1][y][0] ==1}},y[x],{x,0,"8"-1}]
 

\[ y(x)\to \frac {x^7}{896}-\frac {x^6}{320}+\frac {x^4}{16}-\frac {x^3}{4}+x \]