Internal problem ID [13663]
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 (e).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\frac {y^{\prime }}{\left (x -3\right )^{2}}+\frac {y}{\left (x -4\right )^{2}}=0} \] With the expansion point for the power series method at \(x = 4\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 1199
Order:=6; dsolve(diff(y(x),x$2)+1/(x-3)^2*diff(y(x),x)+1/(x-4)^2*y(x)=0,y(x),type='series',x=4);
\[ y \left (x \right ) = \sqrt {x -4}\, \left (c_{2} \left (x -4\right )^{\frac {i \sqrt {3}}{2}} \left (1-\frac {1}{2} \left (x -4\right )+\frac {5 i \sqrt {3}+7}{8 i \sqrt {3}+16} \left (x -4\right )^{2}-\frac {1}{12} \frac {5+36 i \sqrt {3}}{\left (i \sqrt {3}+3\right ) \left (i \sqrt {3}+2\right )} \left (x -4\right )^{3}+\frac {1}{96} \frac {1313 i \sqrt {3}-865}{\left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+3\right ) \left (i \sqrt {3}+2\right )} \left (x -4\right )^{4}-\frac {1}{240} \frac {-23995+15978 i \sqrt {3}}{\left (i \sqrt {3}+5\right ) \left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+3\right ) \left (i \sqrt {3}+2\right )} \left (x -4\right )^{5}+\operatorname {O}\left (\left (x -4\right )^{6}\right )\right )+c_{1} \left (x -4\right )^{-\frac {i \sqrt {3}}{2}} \left (1-\frac {1}{2} \left (x -4\right )+\frac {5 i \sqrt {3}-7}{8 i \sqrt {3}-16} \left (x -4\right )^{2}+\frac {1}{12} \frac {-5+36 i \sqrt {3}}{\left (i \sqrt {3}-3\right ) \left (i \sqrt {3}-2\right )} \left (x -4\right )^{3}+\frac {1}{96} \frac {1313 i \sqrt {3}+865}{\left (i \sqrt {3}-4\right ) \left (i \sqrt {3}-3\right ) \left (i \sqrt {3}-2\right )} \left (x -4\right )^{4}+\frac {1}{240} \frac {23995+15978 i \sqrt {3}}{\left (i \sqrt {3}-5\right ) \left (i \sqrt {3}-4\right ) \left (i \sqrt {3}-3\right ) \left (i \sqrt {3}-2\right )} \left (x -4\right )^{5}+\operatorname {O}\left (\left (x -4\right )^{6}\right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.007 (sec). Leaf size: 2225
AsymptoticDSolveValue[y''[x]+1/(x-3)^2*y'[x]+1/(x-4)^2*y[x]==0,y[x],{x,4,5}]
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