Internal problem ID [2422]
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 11.4. page
758
Problem number: 16.
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 }-\left (x +5\right ) 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: 503
Order:=6; dsolve(x^2*diff(y(x),x$2)+x*(1-x)*diff(y(x),x)-(5+x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{-\sqrt {5}} \left (1+\frac {\sqrt {5}-1}{-1+2 \sqrt {5}} x +\frac {-2+\sqrt {5}}{-4+8 \sqrt {5}} x^{2}+\frac {\left (\sqrt {5}-3\right ) \left (-2+\sqrt {5}\right )}{276-96 \sqrt {5}} x^{3}+\frac {\left (\sqrt {5}-4\right ) \left (\sqrt {5}-3\right )}{2208-768 \sqrt {5}} x^{4}+\frac {\left (-5+\sqrt {5}\right ) \left (\sqrt {5}-4\right ) \left (\sqrt {5}-3\right )}{41280 \sqrt {5}-93600} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} x^{\sqrt {5}} \left (1+\frac {\sqrt {5}+1}{1+2 \sqrt {5}} x +\frac {\sqrt {5}+2}{4+8 \sqrt {5}} x^{2}+\frac {\left (3+\sqrt {5}\right ) \left (\sqrt {5}+2\right )}{276+96 \sqrt {5}} x^{3}+\frac {\left (\sqrt {5}+4\right ) \left (3+\sqrt {5}\right )}{2208+768 \sqrt {5}} x^{4}+\frac {\left (5+\sqrt {5}\right ) \left (\sqrt {5}+4\right ) \left (3+\sqrt {5}\right )}{41280 \sqrt {5}+93600} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 1093
AsymptoticDSolveValue[x^2*y''[x]+x*(1-x)*y'[x]-(5+x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to \left (\frac {\left (-5-\sqrt {5}\right ) \left (-4-\sqrt {5}\right ) \left (-3-\sqrt {5}\right ) \left (-2-\sqrt {5}\right ) \left (1+\sqrt {5}\right ) x^5}{\left (-4+\sqrt {5}+\sqrt {5} \left (1+\sqrt {5}\right )\right ) \left (-3+\sqrt {5}+\left (1+\sqrt {5}\right ) \left (2+\sqrt {5}\right )\right ) \left (-2+\sqrt {5}+\left (2+\sqrt {5}\right ) \left (3+\sqrt {5}\right )\right ) \left (-1+\sqrt {5}+\left (3+\sqrt {5}\right ) \left (4+\sqrt {5}\right )\right ) \left (\sqrt {5}+\left (4+\sqrt {5}\right ) \left (5+\sqrt {5}\right )\right )}-\frac {\left (-4-\sqrt {5}\right ) \left (-3-\sqrt {5}\right ) \left (-2-\sqrt {5}\right ) \left (1+\sqrt {5}\right ) x^4}{\left (-4+\sqrt {5}+\sqrt {5} \left (1+\sqrt {5}\right )\right ) \left (-3+\sqrt {5}+\left (1+\sqrt {5}\right ) \left (2+\sqrt {5}\right )\right ) \left (-2+\sqrt {5}+\left (2+\sqrt {5}\right ) \left (3+\sqrt {5}\right )\right ) \left (-1+\sqrt {5}+\left (3+\sqrt {5}\right ) \left (4+\sqrt {5}\right )\right )}+\frac {\left (-3-\sqrt {5}\right ) \left (-2-\sqrt {5}\right ) \left (1+\sqrt {5}\right ) x^3}{\left (-4+\sqrt {5}+\sqrt {5} \left (1+\sqrt {5}\right )\right ) \left (-3+\sqrt {5}+\left (1+\sqrt {5}\right ) \left (2+\sqrt {5}\right )\right ) \left (-2+\sqrt {5}+\left (2+\sqrt {5}\right ) \left (3+\sqrt {5}\right )\right )}-\frac {\left (-2-\sqrt {5}\right ) \left (1+\sqrt {5}\right ) x^2}{\left (-4+\sqrt {5}+\sqrt {5} \left (1+\sqrt {5}\right )\right ) \left (-3+\sqrt {5}+\left (1+\sqrt {5}\right ) \left (2+\sqrt {5}\right )\right )}+\frac {\left (1+\sqrt {5}\right ) x}{-4+\sqrt {5}+\sqrt {5} \left (1+\sqrt {5}\right )}+1\right ) c_1 x^{\sqrt {5}}+\left (\frac {\left (1-\sqrt {5}\right ) \left (-5+\sqrt {5}\right ) \left (-4+\sqrt {5}\right ) \left (-3+\sqrt {5}\right ) \left (-2+\sqrt {5}\right ) x^5}{\left (-4-\sqrt {5}-\sqrt {5} \left (1-\sqrt {5}\right )\right ) \left (-3-\sqrt {5}+\left (1-\sqrt {5}\right ) \left (2-\sqrt {5}\right )\right ) \left (-2-\sqrt {5}+\left (2-\sqrt {5}\right ) \left (3-\sqrt {5}\right )\right ) \left (-1-\sqrt {5}+\left (3-\sqrt {5}\right ) \left (4-\sqrt {5}\right )\right ) \left (-\sqrt {5}+\left (4-\sqrt {5}\right ) \left (5-\sqrt {5}\right )\right )}-\frac {\left (1-\sqrt {5}\right ) \left (-4+\sqrt {5}\right ) \left (-3+\sqrt {5}\right ) \left (-2+\sqrt {5}\right ) x^4}{\left (-4-\sqrt {5}-\sqrt {5} \left (1-\sqrt {5}\right )\right ) \left (-3-\sqrt {5}+\left (1-\sqrt {5}\right ) \left (2-\sqrt {5}\right )\right ) \left (-2-\sqrt {5}+\left (2-\sqrt {5}\right ) \left (3-\sqrt {5}\right )\right ) \left (-1-\sqrt {5}+\left (3-\sqrt {5}\right ) \left (4-\sqrt {5}\right )\right )}+\frac {\left (1-\sqrt {5}\right ) \left (-3+\sqrt {5}\right ) \left (-2+\sqrt {5}\right ) x^3}{\left (-4-\sqrt {5}-\sqrt {5} \left (1-\sqrt {5}\right )\right ) \left (-3-\sqrt {5}+\left (1-\sqrt {5}\right ) \left (2-\sqrt {5}\right )\right ) \left (-2-\sqrt {5}+\left (2-\sqrt {5}\right ) \left (3-\sqrt {5}\right )\right )}-\frac {\left (1-\sqrt {5}\right ) \left (-2+\sqrt {5}\right ) x^2}{\left (-4-\sqrt {5}-\sqrt {5} \left (1-\sqrt {5}\right )\right ) \left (-3-\sqrt {5}+\left (1-\sqrt {5}\right ) \left (2-\sqrt {5}\right )\right )}+\frac {\left (1-\sqrt {5}\right ) x}{-4-\sqrt {5}-\sqrt {5} \left (1-\sqrt {5}\right )}+1\right ) c_2 x^{-\sqrt {5}} \]