Internal problem ID [1320]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 7 Series Solutions of Linear Second Equations. 7.5 THE METHOD OF FROBENIUS
I. Exercises 7.5. Page 358
Problem number: 31.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} \left (2+x \right ) y^{\prime \prime }+5 x \left (1-x \right ) y^{\prime }-\left (-8 x +2\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 45
Order:=6; dsolve(x^2*(2+x)*diff(y(x),x$2)+5*x*(1-x)*diff(y(x),x)-(2-8*x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{2} x^{\frac {5}{2}} \left (1-\frac {3}{4} x +\frac {5}{96} x^{2}+\frac {5}{4224} x^{3}+\frac {5}{292864} x^{4}-\frac {1}{3514368} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{1} \left (1+8 x +60 x^{2}-160 x^{3}+40 x^{4}+\mathrm {O}\left (x^{6}\right )\right )}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 73
AsymptoticDSolveValue[x^2*(2+x)*y''[x]+5*x*(1-x)*y'[x]-(2-8*x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to \frac {c_2 \left (40 x^4-160 x^3+60 x^2+8 x+1\right )}{x^2}+c_1 \sqrt {x} \left (-\frac {x^5}{3514368}+\frac {5 x^4}{292864}+\frac {5 x^3}{4224}+\frac {5 x^2}{96}-\frac {3 x}{4}+1\right ) \]