Internal problem ID [1374]
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
II. Exercises 7.6. Page 374
Problem number: 22.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {2 x^{2} \left (x +1\right ) y^{\prime \prime }-x \left (6-x \right ) y^{\prime }+\left (8-x \right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.013 (sec). Leaf size: 69
Order:=6; dsolve(2*x^2*(1+x)*diff(y(x),x$2)-x*(6-x)*diff(y(x),x)+(8-x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1-\frac {5}{2} x +\frac {35}{8} x^{2}-\frac {105}{16} x^{3}+\frac {1155}{128} x^{4}-\frac {3003}{256} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (\frac {3}{2} x -\frac {57}{16} x^{2}+\frac {583}{96} x^{3}-\frac {13771}{1536} x^{4}+\frac {187339}{15360} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}\right ) x^{2} \]
✓ Solution by Mathematica
Time used: 0.008 (sec). Leaf size: 128
AsymptoticDSolveValue[2*x^2*(1+x)*y''[x]-x*(6-x)*y'[x]+(8-x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (-\frac {3003 x^5}{256}+\frac {1155 x^4}{128}-\frac {105 x^3}{16}+\frac {35 x^2}{8}-\frac {5 x}{2}+1\right ) x^2+c_2 \left (\left (\frac {187339 x^5}{15360}-\frac {13771 x^4}{1536}+\frac {583 x^3}{96}-\frac {57 x^2}{16}+\frac {3 x}{2}\right ) x^2+\left (-\frac {3003 x^5}{256}+\frac {1155 x^4}{128}-\frac {105 x^3}{16}+\frac {35 x^2}{8}-\frac {5 x}{2}+1\right ) x^2 \log (x)\right ) \]