Internal problem ID [2412]
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: 6.
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 }-7 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: 478
Order:=6; dsolve(x^2*diff(y(x),x$2)+x*(1-x)*diff(y(x),x)-7*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{-\sqrt {7}} \left (1+\frac {\sqrt {7}}{-1+2 \sqrt {7}} x +\frac {\sqrt {7}}{-4+8 \sqrt {7}} x^{2}+\frac {\left (\sqrt {7}-2\right ) \sqrt {7}}{372-96 \sqrt {7}} x^{3}+\frac {\sqrt {7}\, \left (\sqrt {7}-3\right )}{2976-768 \sqrt {7}} x^{4}+\frac {\left (\sqrt {7}-4\right ) \left (\sqrt {7}-3\right ) \sqrt {7}}{48960 \sqrt {7}-128160} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} x^{\sqrt {7}} \left (1+\frac {\sqrt {7}}{1+2 \sqrt {7}} x +\frac {\sqrt {7}}{4+8 \sqrt {7}} x^{2}+\frac {\left (\sqrt {7}+2\right ) \sqrt {7}}{372+96 \sqrt {7}} x^{3}+\frac {\left (\sqrt {7}+3\right ) \sqrt {7}}{2976+768 \sqrt {7}} x^{4}+\frac {\left (\sqrt {7}+4\right ) \left (\sqrt {7}+3\right ) \sqrt {7}}{48960 \sqrt {7}+128160} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 1066
AsymptoticDSolveValue[x^2*y''[x]+x*(1-x)*y'[x]-7*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to \left (\frac {\sqrt {7} \left (1+\sqrt {7}\right ) \left (2+\sqrt {7}\right ) \left (3+\sqrt {7}\right ) \left (4+\sqrt {7}\right ) x^5}{\left (-6+\sqrt {7}+\sqrt {7} \left (1+\sqrt {7}\right )\right ) \left (-5+\sqrt {7}+\left (1+\sqrt {7}\right ) \left (2+\sqrt {7}\right )\right ) \left (-4+\sqrt {7}+\left (2+\sqrt {7}\right ) \left (3+\sqrt {7}\right )\right ) \left (-3+\sqrt {7}+\left (3+\sqrt {7}\right ) \left (4+\sqrt {7}\right )\right ) \left (-2+\sqrt {7}+\left (4+\sqrt {7}\right ) \left (5+\sqrt {7}\right )\right )}+\frac {\sqrt {7} \left (1+\sqrt {7}\right ) \left (2+\sqrt {7}\right ) \left (3+\sqrt {7}\right ) x^4}{\left (-6+\sqrt {7}+\sqrt {7} \left (1+\sqrt {7}\right )\right ) \left (-5+\sqrt {7}+\left (1+\sqrt {7}\right ) \left (2+\sqrt {7}\right )\right ) \left (-4+\sqrt {7}+\left (2+\sqrt {7}\right ) \left (3+\sqrt {7}\right )\right ) \left (-3+\sqrt {7}+\left (3+\sqrt {7}\right ) \left (4+\sqrt {7}\right )\right )}+\frac {\sqrt {7} \left (1+\sqrt {7}\right ) \left (2+\sqrt {7}\right ) x^3}{\left (-6+\sqrt {7}+\sqrt {7} \left (1+\sqrt {7}\right )\right ) \left (-5+\sqrt {7}+\left (1+\sqrt {7}\right ) \left (2+\sqrt {7}\right )\right ) \left (-4+\sqrt {7}+\left (2+\sqrt {7}\right ) \left (3+\sqrt {7}\right )\right )}+\frac {\sqrt {7} \left (1+\sqrt {7}\right ) x^2}{\left (-6+\sqrt {7}+\sqrt {7} \left (1+\sqrt {7}\right )\right ) \left (-5+\sqrt {7}+\left (1+\sqrt {7}\right ) \left (2+\sqrt {7}\right )\right )}+\frac {\sqrt {7} x}{-6+\sqrt {7}+\sqrt {7} \left (1+\sqrt {7}\right )}+1\right ) c_1 x^{\sqrt {7}}+\left (-\frac {\sqrt {7} \left (1-\sqrt {7}\right ) \left (2-\sqrt {7}\right ) \left (3-\sqrt {7}\right ) \left (4-\sqrt {7}\right ) x^5}{\left (-6-\sqrt {7}-\sqrt {7} \left (1-\sqrt {7}\right )\right ) \left (-5-\sqrt {7}+\left (1-\sqrt {7}\right ) \left (2-\sqrt {7}\right )\right ) \left (-4-\sqrt {7}+\left (2-\sqrt {7}\right ) \left (3-\sqrt {7}\right )\right ) \left (-3-\sqrt {7}+\left (3-\sqrt {7}\right ) \left (4-\sqrt {7}\right )\right ) \left (-2-\sqrt {7}+\left (4-\sqrt {7}\right ) \left (5-\sqrt {7}\right )\right )}-\frac {\sqrt {7} \left (1-\sqrt {7}\right ) \left (2-\sqrt {7}\right ) \left (3-\sqrt {7}\right ) x^4}{\left (-6-\sqrt {7}-\sqrt {7} \left (1-\sqrt {7}\right )\right ) \left (-5-\sqrt {7}+\left (1-\sqrt {7}\right ) \left (2-\sqrt {7}\right )\right ) \left (-4-\sqrt {7}+\left (2-\sqrt {7}\right ) \left (3-\sqrt {7}\right )\right ) \left (-3-\sqrt {7}+\left (3-\sqrt {7}\right ) \left (4-\sqrt {7}\right )\right )}-\frac {\sqrt {7} \left (1-\sqrt {7}\right ) \left (2-\sqrt {7}\right ) x^3}{\left (-6-\sqrt {7}-\sqrt {7} \left (1-\sqrt {7}\right )\right ) \left (-5-\sqrt {7}+\left (1-\sqrt {7}\right ) \left (2-\sqrt {7}\right )\right ) \left (-4-\sqrt {7}+\left (2-\sqrt {7}\right ) \left (3-\sqrt {7}\right )\right )}-\frac {\sqrt {7} \left (1-\sqrt {7}\right ) x^2}{\left (-6-\sqrt {7}-\sqrt {7} \left (1-\sqrt {7}\right )\right ) \left (-5-\sqrt {7}+\left (1-\sqrt {7}\right ) \left (2-\sqrt {7}\right )\right )}-\frac {\sqrt {7} x}{-6-\sqrt {7}-\sqrt {7} \left (1-\sqrt {7}\right )}+1\right ) c_2 x^{-\sqrt {7}} \]