Internal problem ID [4721]
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(iii), 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 (x^{2}+1\right ) y^{\prime \prime \prime }-\left (4 x^{2}+2\right ) x^{2} y^{\prime \prime }+\left (10 x^{2}+4\right ) x y^{\prime }-\left (12 x^{2}+4\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.026 (sec). Leaf size: 52
Order:=6; dsolve(x^3*(1+x^2)*diff(y(x),x$3)-(2+4*x^2)*x^2*diff(y(x),x$2)+(4+10*x^2)*x*diff(y(x),x)-(4+12*x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = x \left (\left (2+2 x^{2}+\mathrm {O}\left (x^{6}\right )\right ) c_{3}+x \left (\left (1+\mathrm {O}\left (x^{6}\right )\right ) c_{1}+c_{2} \left (\ln \relax (x ) \left (2+\mathrm {O}\left (x^{6}\right )\right )+\left (5+\mathrm {O}\left (x^{6}\right )\right )\right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.036 (sec). Leaf size: 30
AsymptoticDSolveValue[x^3*(1+x^2)*y'''[x]-(2+4*x^2)*x^2*y''[x]+(4+10*x^2)*x*y'[x]-(4+12*x^2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (2 x^3+2 x\right )+c_2 x^2+c_3 x^2 \log (x) \]