Internal problem ID [11920]
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: 19.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +y \left (x -1\right )=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 63
Order:=6; dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)+(x-1)*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \frac {c_{1} x^{2} \left (1-\frac {1}{3} x +\frac {1}{24} x^{2}-\frac {1}{360} x^{3}+\frac {1}{8640} x^{4}-\frac {1}{302400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (x^{2}-\frac {1}{3} x^{3}+\frac {1}{24} x^{4}-\frac {1}{360} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2-2 x +\frac {4}{9} x^{3}-\frac {25}{288} x^{4}+\frac {157}{21600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.017 (sec). Leaf size: 83
AsymptoticDSolveValue[x^2*y''[x]+x*y'[x]+(x-1)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {31 x^4-176 x^3+144 x^2+576 x+576}{576 x}-\frac {1}{48} x \left (x^2-8 x+24\right ) \log (x)\right )+c_2 \left (\frac {x^5}{8640}-\frac {x^4}{360}+\frac {x^3}{24}-\frac {x^2}{3}+x\right ) \]