Internal problem ID [11905]
Book: Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi.
2004.
Section: Chapter 6, Series solutions of linear differential equations. Section 6.2 (Frobenius).
Exercises page 251
Problem number: 4.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (x^{5}+x^{4}-6 x^{3}\right ) y^{\prime \prime }+x^{2} y^{\prime }+y \left (x -2\right )=0} \] With the expansion point for the power series method at \(x = 0\).
✗ Solution by Maple
Order:=6; dsolve((x^5+x^4-6*x^3)*diff(y(x),x$2)+x^2*diff(y(x),x)+(x-2)*y(x)=0,y(x),type='series',x=0);
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 0.226 (sec). Leaf size: 282
AsymptoticDSolveValue[(x^5+x^4-6*x^3)*y''[x]+x^2*y'[x]+(x-2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 e^{-\frac {2 i}{\sqrt {3} \sqrt {x}}} x^{5/6} \left (-\frac {70670717962217 i x^{9/2}}{8463329722368 \sqrt {3}}+\frac {454703707 i x^{7/2}}{544195584 \sqrt {3}}-\frac {287057 i x^{5/2}}{1679616 \sqrt {3}}+\frac {22 i x^{3/2}}{243 \sqrt {3}}+\frac {28128149072197063 x^5}{1523399350026240}-\frac {222818846149 x^4}{156728328192}+\frac {35197783 x^3}{181398528}-\frac {14123 x^2}{279936}+\frac {17 x}{216}-\frac {7 i \sqrt {x}}{6 \sqrt {3}}+1\right )+c_2 e^{\frac {2 i}{\sqrt {3} \sqrt {x}}} x^{5/6} \left (\frac {70670717962217 i x^{9/2}}{8463329722368 \sqrt {3}}-\frac {454703707 i x^{7/2}}{544195584 \sqrt {3}}+\frac {287057 i x^{5/2}}{1679616 \sqrt {3}}-\frac {22 i x^{3/2}}{243 \sqrt {3}}+\frac {28128149072197063 x^5}{1523399350026240}-\frac {222818846149 x^4}{156728328192}+\frac {35197783 x^3}{181398528}-\frac {14123 x^2}{279936}+\frac {17 x}{216}+\frac {7 i \sqrt {x}}{6 \sqrt {3}}+1\right ) \]