46.2.9 problem 10

Internal problem ID [7312]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 5. Series Solutions of ODEs. Special Functions. Problem set 5.3. Extended Power Series Method: Frobenius Method page 186
Problem number : 10
Date solved : Monday, January 27, 2025 at 02:50:09 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+2 y^{\prime }+4 y x&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 32

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

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

✓ Solution by Mathematica

Time used: 0.010 (sec). Leaf size: 40

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

\[ y(x)\to c_1 \left (\frac {2 x^3}{3}-2 x+\frac {1}{x}\right )+c_2 \left (\frac {2 x^4}{15}-\frac {2 x^2}{3}+1\right ) \]