Internal problem ID [1429]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF FROBENIUS
III. Exercises 7.7. Page 389
Problem number: 13.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
Solve \begin {gather*} \boxed {x \left (x +1\right ) y^{\prime \prime }-4 y^{\prime }-2 y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 40
Order:=6; dsolve(x*(1+x)*diff(y(x),x$2)-4*diff(y(x),x)-2*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{5} \left (1-3 x +6 x^{2}-10 x^{3}+15 x^{4}-21 x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (2880-1440 x +480 x^{2}-480 x^{5}+\mathrm {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.041 (sec). Leaf size: 48
AsymptoticDSolveValue[x*(1+x)*y''[x]-4*y'[x]-2*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {x^2}{6}-\frac {x}{2}+1\right )+c_2 \left (15 x^9-10 x^8+6 x^7-3 x^6+x^5\right ) \]