Internal problem ID [4893]
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: 12.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+6 x y^{\prime }+\left (4 x^{2}+6\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 35
Order:=6; dsolve(x^2*diff(y(x),x$2)+6*x*diff(y(x),x)+(4*x^2+6)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{1} \left (1-\frac {2}{3} x^{2}+\frac {2}{15} x^{4}+\mathrm {O}\left (x^{6}\right )\right ) x +c_{2} \left (1-2 x^{2}+\frac {2}{3} x^{4}+\mathrm {O}\left (x^{6}\right )\right )}{x^{3}} \]
✓ Solution by Mathematica
Time used: 0.015 (sec). Leaf size: 38
AsymptoticDSolveValue[x^2*y''[x]+6*x*y'[x]+(4*x^2+6)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {1}{x^3}+\frac {2 x}{3}-\frac {2}{x}\right )+c_2 \left (\frac {2 x^2}{15}+\frac {1}{x^2}-\frac {2}{3}\right ) \]