Internal problem ID [1379]
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: 27.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} \left (1-x \right ) y^{\prime \prime }+x \left (7+x \right ) y^{\prime }+\left (9-x \right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.022 (sec). Leaf size: 63
Order:=8; dsolve(x^2*(1-x)*diff(y(x),x$2)+x*(7+x)*diff(y(x),x)+(9-x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1+16 x +36 x^{2}+16 x^{3}+x^{4}+\mathrm {O}\left (x^{8}\right )\right )+\left (\left (-40\right ) x -150 x^{2}-\frac {280}{3} x^{3}-\frac {25}{3} x^{4}+\mathrm {O}\left (x^{8}\right )\right ) c_{2}}{x^{3}} \]
✓ Solution by Mathematica
Time used: 0.014 (sec). Leaf size: 83
AsymptoticDSolveValue[x^2*(1-x)*y''[x]+x*(7+x)*y'[x]+(9-x)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to \frac {c_1 \left (x^4+16 x^3+36 x^2+16 x+1\right )}{x^3}+c_2 \left (\frac {-\frac {25 x^4}{3}-\frac {280 x^3}{3}-150 x^2-40 x}{x^3}+\frac {\left (x^4+16 x^3+36 x^2+16 x+1\right ) \log (x)}{x^3}\right ) \]