Internal problem ID [4887]
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: 6.
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 x^{3} y^{\prime }+\left (x^{2}-2\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.024 (sec). Leaf size: 58
Order:=6; dsolve(x*diff(y(x),x$2)+2*x^3*diff(y(x),x)+(x^2-2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x \left (1+x +\frac {1}{3} x^{2}-\frac {7}{36} x^{3}-\frac {97}{360} x^{4}-\frac {517}{5400} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \relax (x ) \left (2 x +2 x^{2}+\frac {2}{3} x^{3}-\frac {7}{18} x^{4}-\frac {97}{180} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (1-3 x^{2}-\frac {31}{18} x^{3}-\frac {85}{216} x^{4}+\frac {4067}{5400} x^{5}+\mathrm {O}\left (x^{6}\right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.018 (sec). Leaf size: 83
AsymptoticDSolveValue[x*y''[x]+2*x^3*y'[x]+(x^2-2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {1}{216} \left (-x^4-516 x^3-1080 x^2-432 x+216\right )-\frac {1}{18} x \left (7 x^3-12 x^2-36 x-36\right ) \log (x)\right )+c_2 \left (-\frac {97 x^5}{360}-\frac {7 x^4}{36}+\frac {x^3}{3}+x^2+x\right ) \]