Internal problem ID [4767]
Book: Notes on Diffy Qs. Differential Equations for Engineers. By by Jiri Lebl, 2013.
Section: Chapter 7. POWER SERIES METHODS. 7.3.2 The method of Frobenius. Exercises. page
300
Problem number: 7.3.8 (a).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+y x=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: 58
Order:=6; dsolve(x^2*(1+x^2)*diff(y(x),x$2)+x*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x \left (1-\frac {1}{2} x +\frac {1}{12} x^{2}+\frac {11}{144} x^{3}-\frac {83}{2880} x^{4}-\frac {2557}{86400} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \relax (x ) \left (-x +\frac {1}{2} x^{2}-\frac {1}{12} x^{3}-\frac {11}{144} x^{4}+\frac {83}{2880} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (1-\frac {3}{4} x^{2}+\frac {13}{36} x^{3}+\frac {25}{1728} x^{4}-\frac {8743}{86400} x^{5}+\mathrm {O}\left (x^{6}\right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.037 (sec). Leaf size: 87
AsymptoticDSolveValue[x^2*(1+x^2)*y''[x]+x*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {157 x^4+768 x^3-2160 x^2+1728 x+1728}{1728}-\frac {1}{144} x \left (11 x^3+12 x^2-72 x+144\right ) \log (x)\right )+c_2 \left (-\frac {83 x^5}{2880}+\frac {11 x^4}{144}+\frac {x^3}{12}-\frac {x^2}{2}+x\right ) \]