Internal problem ID [4718]
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. 5, page 256.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x \left (-x^{2}+2\right ) y^{\prime \prime }-\left (x^{2}+4 x +2\right ) \left (\left (1-x \right ) y^{\prime }+y\right )=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.029 (sec). Leaf size: 44
Order:=6; dsolve(x*(2-x^2)*diff(y(x),x$2)-(x^2+4*x+2)*((1-x)*diff(y(x),x)+y(x))=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{2} \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (-2+2 x +4 x^{2}+4 x^{3}+2 x^{4}+\frac {2}{3} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.039 (sec). Leaf size: 64
AsymptoticDSolveValue[x*(2-x^2)*y''[x]-(x^2+4*x+2)*((1-x)*y'[x]+y[x])==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (-\frac {5 x^4}{4}-\frac {5 x^3}{2}-\frac {5 x^2}{2}-x+1\right )+c_2 \left (\frac {x^6}{24}+\frac {x^5}{6}+\frac {x^4}{2}+x^3+x^2\right ) \]