Internal problem ID [4720]
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(ii), page 257.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{3} \left (1+x \right ) y^{\prime \prime \prime }-\left (2+4 x \right ) x^{2} y^{\prime \prime }+\left (4+10 x \right ) x y^{\prime }-\left (4+12 x \right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 79
Order:=6; dsolve(x^3*(1+x)*diff(y(x),x$3)-(2+4*x)*x^2*diff(y(x),x$2)+(4+10*x)*x*diff(y(x),x)-(4+12*x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = x \left (\ln \relax (x )^{2} \left (2 x +\mathrm {O}\left (x^{6}\right )\right ) c_{3}+\ln \relax (x ) \left (2+\mathrm {O}\left (x^{6}\right )\right ) c_{2} x +c_{1} x \left (1+\mathrm {O}\left (x^{6}\right )\right )+2 \ln \relax (x ) \left (\left (-4\right ) x +\mathrm {O}\left (x^{6}\right )\right ) c_{3}+\left (5+\mathrm {O}\left (x^{6}\right )\right ) c_{2} x +\left (2+4 x +2 x^{2}+\mathrm {O}\left (x^{6}\right )\right ) c_{3}\right ) \]
✓ Solution by Mathematica
Time used: 0.778 (sec). Leaf size: 49
AsymptoticDSolveValue[x^3*(1+x)*y'''[x]-(2+4*x)*x^2*y''[x]+(4+10*x)*x*y'[x]-(4+12*x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 x^2+c_1 \left (2 \left (x^2+11 x+1\right ) x+2 x^2 \log ^2(x)-14 x^2 \log (x)\right )+c_3 x^2 \log (x) \]