Internal problem ID [1296]
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: 2.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, _with_symmetry_[0,F(x)]]]
Solve \begin {gather*} \boxed {3 x^{2} y^{\prime \prime }+2 x \left (-2 x^{2}+x +1\right ) y^{\prime }+\left (-8 x^{2}+2 x \right ) 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: 44
Order:=6; dsolve(3*x^2*diff(y(x),x$2)+2*x*(1+x-2*x^2)*diff(y(x),x)+(2*x-8*x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{\frac {1}{3}} \left (1-\frac {2}{3} x +\frac {8}{9} x^{2}-\frac {40}{81} x^{3}+\frac {92}{243} x^{4}-\frac {664}{3645} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (1-x +\frac {6}{5} x^{2}-\frac {4}{5} x^{3}+\frac {32}{55} x^{4}-\frac {24}{77} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 83
AsymptoticDSolveValue[3*x^2*y''[x]+2*x*(1+x-2*x^2)*y'[x]+(2*x-8*x^2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \sqrt [3]{x} \left (-\frac {664 x^5}{3645}+\frac {92 x^4}{243}-\frac {40 x^3}{81}+\frac {8 x^2}{9}-\frac {2 x}{3}+1\right )+c_2 \left (-\frac {24 x^5}{77}+\frac {32 x^4}{55}-\frac {4 x^3}{5}+\frac {6 x^2}{5}-x+1\right ) \]