Internal problem ID [5476]
Book: A treatise on Differential Equations by A. R. Forsyth. 6th edition. 1929. Macmillan Co. ltd.
New York, reprinted 1956
Section: Chapter VI. Note I. Integration of linear equations in series by the method of Frobenius.
page 243
Problem number: Ex. 6(v), page 257.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (1-x \right ) x^{2} y^{\prime \prime }+\left (5 x -4\right ) x y^{\prime }+\left (6-9 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 43
Order:=6; dsolve((1-x)*x^2*diff(y(x),x$2)+(5*x-4)*x*diff(y(x),x)+(6-9*x)*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = x^{2} \left (c_{1} x \left (1+\operatorname {O}\left (x^{6}\right )\right )+\left (x +\operatorname {O}\left (x^{6}\right )\right ) \ln \left (x \right ) c_{2} +\left (1-x +\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \]
✓ Solution by Mathematica
Time used: 0.05 (sec). Leaf size: 30
AsymptoticDSolveValue[(1-x)*x^2*y''[x]+(5*x-4)*x*y'[x]+(6-9*x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 x^3+c_1 \left (x^3 \log (x)-x^2 (3 x-1)\right ) \]