Internal problem ID [4722]
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(iv), page 257.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {2 \left (2-x \right ) x^{2} y^{\prime \prime }-\left (4-x \right ) x y^{\prime }+\left (3-x \right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 47
Order:=6; dsolve(2*(2-x)*x^2*diff(y(x),x$2)-(4-x)*x*diff(y(x),x)+(3-x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \sqrt {x}\, \left (x \left (1+\frac {1}{8} x +\frac {1}{32} x^{2}+\frac {5}{512} x^{3}+\frac {7}{2048} x^{4}+\frac {21}{16384} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) c_{1}+\left (1+\frac {1}{4} x +\frac {1}{32} x^{2}+\frac {1}{128} x^{3}+\frac {5}{2048} x^{4}+\frac {7}{8192} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.082 (sec). Leaf size: 94
AsymptoticDSolveValue[2*(2-x)*x^2*y''[x]-(4-x)*x*y'[x]+(3-x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (-\frac {5 x^{9/2}}{2048}-\frac {x^{7/2}}{128}-\frac {x^{5/2}}{32}-\frac {x^{3/2}}{4}+\sqrt {x}\right )+c_2 \left (\frac {7 x^{11/2}}{2048}+\frac {5 x^{9/2}}{512}+\frac {x^{7/2}}{32}+\frac {x^{5/2}}{8}+x^{3/2}\right ) \]