Internal problem ID [5470]
Book: Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres.
McGraw Hill 1952
Section: Chapter 27. The Legendre, Bessel and Gauss Equations. Supplemetary problems. Page
230
Problem number: 20.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {z^{\prime \prime }+t z^{\prime }+\left (t^{2}-\frac {1}{9}\right ) z=0} \] With the expansion point for the power series method at \(t = 0\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 34
Order:=6; dsolve(diff(z(t),t$2)+t*diff(z(t),t)+(t^2-1/9)*z(t)=0,z(t),type='series',t=0);
\[ z \left (t \right ) = \left (1+\frac {1}{18} t^{2}-\frac {179}{1944} t^{4}\right ) z \left (0\right )+\left (t -\frac {4}{27} t^{3}-\frac {139}{4860} t^{5}\right ) D\left (z \right )\left (0\right )+O\left (t^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 42
AsymptoticDSolveValue[z''[t]+t*z'[t]+(t^2-1/9)*z[t]==0,z[t],{t,0,5}]
\[ z(t)\to c_2 \left (-\frac {139 t^5}{4860}-\frac {4 t^3}{27}+t\right )+c_1 \left (-\frac {179 t^4}{1944}+\frac {t^2}{18}+1\right ) \]