Internal problem ID [2418]
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: 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 }+y^{\prime } x -\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: 0.015 (sec). Leaf size: 321
Order:=6; dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)-(2+x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{-\sqrt {2}} \left (1-\frac {1}{-1+2 \sqrt {2}} x +\frac {1}{20-12 \sqrt {2}} x^{2}-\frac {1}{228 \sqrt {2}-324} x^{3}+\frac {1}{8832-6240 \sqrt {2}} x^{4}-\frac {1}{244320 \sqrt {2}-345600} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} x^{\sqrt {2}} \left (1+\frac {1}{1+2 \sqrt {2}} x +\frac {1}{20+12 \sqrt {2}} x^{2}+\frac {1}{228 \sqrt {2}+324} x^{3}+\frac {1}{8832+6240 \sqrt {2}} x^{4}+\frac {1}{244320 \sqrt {2}+345600} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 843
AsymptoticDSolveValue[x^2*y''[x]+x*y'[x]-(2+x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to \left (\frac {x^5}{\left (-1+\sqrt {2}+\sqrt {2} \left (1+\sqrt {2}\right )\right ) \left (\sqrt {2}+\left (1+\sqrt {2}\right ) \left (2+\sqrt {2}\right )\right ) \left (1+\sqrt {2}+\left (2+\sqrt {2}\right ) \left (3+\sqrt {2}\right )\right ) \left (2+\sqrt {2}+\left (3+\sqrt {2}\right ) \left (4+\sqrt {2}\right )\right ) \left (3+\sqrt {2}+\left (4+\sqrt {2}\right ) \left (5+\sqrt {2}\right )\right )}+\frac {x^4}{\left (-1+\sqrt {2}+\sqrt {2} \left (1+\sqrt {2}\right )\right ) \left (\sqrt {2}+\left (1+\sqrt {2}\right ) \left (2+\sqrt {2}\right )\right ) \left (1+\sqrt {2}+\left (2+\sqrt {2}\right ) \left (3+\sqrt {2}\right )\right ) \left (2+\sqrt {2}+\left (3+\sqrt {2}\right ) \left (4+\sqrt {2}\right )\right )}+\frac {x^3}{\left (-1+\sqrt {2}+\sqrt {2} \left (1+\sqrt {2}\right )\right ) \left (\sqrt {2}+\left (1+\sqrt {2}\right ) \left (2+\sqrt {2}\right )\right ) \left (1+\sqrt {2}+\left (2+\sqrt {2}\right ) \left (3+\sqrt {2}\right )\right )}+\frac {x^2}{\left (-1+\sqrt {2}+\sqrt {2} \left (1+\sqrt {2}\right )\right ) \left (\sqrt {2}+\left (1+\sqrt {2}\right ) \left (2+\sqrt {2}\right )\right )}+\frac {x}{-1+\sqrt {2}+\sqrt {2} \left (1+\sqrt {2}\right )}+1\right ) c_1 x^{\sqrt {2}}+\left (\frac {x^5}{\left (-1-\sqrt {2}-\sqrt {2} \left (1-\sqrt {2}\right )\right ) \left (-\sqrt {2}+\left (1-\sqrt {2}\right ) \left (2-\sqrt {2}\right )\right ) \left (1-\sqrt {2}+\left (2-\sqrt {2}\right ) \left (3-\sqrt {2}\right )\right ) \left (2-\sqrt {2}+\left (3-\sqrt {2}\right ) \left (4-\sqrt {2}\right )\right ) \left (3-\sqrt {2}+\left (4-\sqrt {2}\right ) \left (5-\sqrt {2}\right )\right )}+\frac {x^4}{\left (-1-\sqrt {2}-\sqrt {2} \left (1-\sqrt {2}\right )\right ) \left (-\sqrt {2}+\left (1-\sqrt {2}\right ) \left (2-\sqrt {2}\right )\right ) \left (1-\sqrt {2}+\left (2-\sqrt {2}\right ) \left (3-\sqrt {2}\right )\right ) \left (2-\sqrt {2}+\left (3-\sqrt {2}\right ) \left (4-\sqrt {2}\right )\right )}+\frac {x^3}{\left (-1-\sqrt {2}-\sqrt {2} \left (1-\sqrt {2}\right )\right ) \left (-\sqrt {2}+\left (1-\sqrt {2}\right ) \left (2-\sqrt {2}\right )\right ) \left (1-\sqrt {2}+\left (2-\sqrt {2}\right ) \left (3-\sqrt {2}\right )\right )}+\frac {x^2}{\left (-1-\sqrt {2}-\sqrt {2} \left (1-\sqrt {2}\right )\right ) \left (-\sqrt {2}+\left (1-\sqrt {2}\right ) \left (2-\sqrt {2}\right )\right )}+\frac {x}{-1-\sqrt {2}-\sqrt {2} \left (1-\sqrt {2}\right )}+1\right ) c_2 x^{-\sqrt {2}} \]