Internal problem ID [1199]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 7 Series Solutions of Linear Second Equations. 7.1 Exercises. Page 318
Problem number: 21.
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 (x +4\right ) y^{\prime }+\left (2-x \right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 55
Order:=6; dsolve(x^2*(1-x)*diff(y(x),x$2)+x*(4+x)*diff(y(x),x)+(2-x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {\ln \relax (x ) \left (9 x +18 x^{2}+3 x^{3}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}+c_{1} \left (1+2 x +\frac {1}{3} x^{2}+\mathrm {O}\left (x^{6}\right )\right ) x +\left (1-5 x -55 x^{2}-\frac {53}{3} x^{3}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.049 (sec). Leaf size: 56
AsymptoticDSolveValue[x^2*(1-x)*y''[x]+x*(4+x)*y'[x]+(2-x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {3 \left (x^2+6 x+3\right ) \log (x)}{x}-\frac {21 x^3+75 x^2+15 x-1}{x^2}\right )+c_2 \left (\frac {x}{3}+\frac {1}{x}+2\right ) \]