Internal problem ID [11912]
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: 11.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {3 x y^{\prime \prime }-\left (x -2\right ) y^{\prime }-2 y=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 44
Order:=6; dsolve(3*x*diff(y(x),x$2)-(x-2)*diff(y(x),x)-2*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = c_{1} x^{\frac {1}{3}} \left (1+\frac {7}{12} x +\frac {5}{36} x^{2}+\frac {13}{648} x^{3}+\frac {1}{486} x^{4}+\frac {19}{116640} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+x +\frac {3}{10} x^{2}+\frac {1}{20} x^{3}+\frac {1}{176} x^{4}+\frac {3}{6160} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 81
AsymptoticDSolveValue[3*x*y''[x]-(x-2)*y'[x]-2*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \sqrt [3]{x} \left (\frac {19 x^5}{116640}+\frac {x^4}{486}+\frac {13 x^3}{648}+\frac {5 x^2}{36}+\frac {7 x}{12}+1\right )+c_2 \left (\frac {3 x^5}{6160}+\frac {x^4}{176}+\frac {x^3}{20}+\frac {3 x^2}{10}+x+1\right ) \]