Internal problem ID [2448]
Book: Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition,
2015
Section: Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.5. page
771
Problem number: 15.
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-x \right ) y^{\prime }-y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.014 (sec). Leaf size: 45
Order:=6; dsolve(x^2*diff(y(x),x$2)+x*(1-x)*diff(y(x),x)-y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x \left (1+\frac {1}{3} x +\frac {1}{12} x^{2}+\frac {1}{60} x^{3}+\frac {1}{360} x^{4}+\frac {1}{2520} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (-2-2 x -x^{2}-\frac {1}{3} x^{3}-\frac {1}{12} x^{4}-\frac {1}{60} x^{5}+\mathrm {O}\left (x^{6}\right )\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.017 (sec). Leaf size: 64
AsymptoticDSolveValue[x^2*y''[x]+x*(1-x)*y'[x]-y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {x^3}{24}+\frac {x^2}{6}+\frac {x}{2}+\frac {1}{x}+1\right )+c_2 \left (\frac {x^5}{360}+\frac {x^4}{60}+\frac {x^3}{12}+\frac {x^2}{3}+x\right ) \]