Internal problem ID [1428]
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: 12.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+x \left (1-2 x \right ) y^{\prime }-\left (x +4\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: 61
Order:=6; dsolve(x^2*diff(y(x),x$2)+x*(1-2*x)*diff(y(x),x)-(4+x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{1} x^{4} \left (1+x +\frac {7}{12} x^{2}+\frac {1}{4} x^{3}+\frac {11}{128} x^{4}+\frac {143}{5760} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \relax (x ) \left (9 x^{4}+9 x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (-144-144 x -36 x^{2}+12 x^{3}-\frac {36}{5} x^{5}+\mathrm {O}\left (x^{6}\right )\right )\right )}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.031 (sec). Leaf size: 75
AsymptoticDSolveValue[x^2*y''[x]+x*(1-2*x)*y'[x]-(4+x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {3 x^4-16 x^3+48 x^2+192 x+192}{192 x^2}-\frac {1}{16} x^2 \log (x)\right )+c_2 \left (\frac {11 x^6}{128}+\frac {x^5}{4}+\frac {7 x^4}{12}+x^3+x^2\right ) \]