Internal problem ID [4891]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x y^{\prime \prime }+2 y^{\prime }+4 x y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.023 (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);
\[ y \relax (x ) = c_{1} \left (1-\frac {2}{3} x^{2}+\frac {2}{15} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (1-2 x^{2}+\frac {2}{3} x^{4}+\mathrm {O}\left (x^{6}\right )\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 40
AsymptoticDSolveValue[x*y''[x]+2*y'[x]+4*x*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {2 x^3}{3}-2 x+\frac {1}{x}\right )+c_2 \left (\frac {2 x^4}{15}-\frac {2 x^2}{3}+1\right ) \]