Internal problem ID [13659]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 35. Modified Power series solutions and basic method of Frobenius. Additional
Exercises. page 715
Problem number: 35.3 (a).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+\frac {x y^{\prime }}{x -2}+\frac {2 y}{x +2}=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 1283
Order:=6; dsolve(x^2*diff(y(x),x$2)+x/(x-2)*diff(y(x),x)+2/(x+2)*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = x^{\frac {3}{4}} \left (c_{2} x^{\frac {i \sqrt {7}}{4}} \left (1+\frac {i \sqrt {7}+11}{8 i \sqrt {7}+16} x +\frac {1}{64} \frac {7 i \sqrt {7}+45}{\left (2+i \sqrt {7}\right ) \left (i \sqrt {7}+4\right )} x^{2}+\frac {1}{256} \frac {223 i \sqrt {7}-43}{\left (2+i \sqrt {7}\right ) \left (i \sqrt {7}+4\right ) \left (i \sqrt {7}+6\right )} x^{3}+\frac {1}{4096} \frac {7577 i \sqrt {7}+979}{\left (2+i \sqrt {7}\right ) \left (i \sqrt {7}+4\right ) \left (i \sqrt {7}+6\right ) \left (i \sqrt {7}+8\right )} x^{4}+\frac {1}{81920} \frac {553875 i \sqrt {7}-1249007}{\left (2+i \sqrt {7}\right ) \left (i \sqrt {7}+4\right ) \left (i \sqrt {7}+6\right ) \left (i \sqrt {7}+8\right ) \left (i \sqrt {7}+10\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} x^{-\frac {i \sqrt {7}}{4}} \left (1+\frac {i \sqrt {7}-11}{8 i \sqrt {7}-16} x -\frac {1}{64} \frac {7 i \sqrt {7}-45}{\left (i \sqrt {7}-2\right ) \left (i \sqrt {7}-4\right )} x^{2}+\frac {1}{256} \frac {223 i \sqrt {7}+43}{\left (i \sqrt {7}-2\right ) \left (i \sqrt {7}-4\right ) \left (i \sqrt {7}-6\right )} x^{3}-\frac {1}{4096} \frac {7577 i \sqrt {7}-979}{\left (i \sqrt {7}-2\right ) \left (i \sqrt {7}-4\right ) \left (i \sqrt {7}-6\right ) \left (i \sqrt {7}-8\right )} x^{4}+\frac {1}{81920} \frac {553875 i \sqrt {7}+1249007}{\left (i \sqrt {7}-2\right ) \left (i \sqrt {7}-4\right ) \left (i \sqrt {7}-6\right ) \left (i \sqrt {7}-8\right ) \left (i \sqrt {7}-10\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 6290
AsymptoticDSolveValue[x^2*y''[x]+x/(x-2)*y'[x]+2/(x+2)*y[x]==0,y[x],{x,0,5}]
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