Internal problem ID [1819]
Book: Differential equations and their applications, 3rd ed., M. Braun
Section: Section 2.8.2, Regular singular points, the method of Frobenius. Page 214
Problem number: 29.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {t^{2} y^{\prime \prime }+t y^{\prime }+\left (t +1\right ) y=0} \end {gather*} With the expansion point for the power series method at \(t = 0\).
✓ Solution by Maple
Time used: 0.01 (sec). Leaf size: 69
Order:=6; dsolve(t^2*diff(y(t),t$2)+t*diff(y(t),t)+(1+t)*y(t)=0,y(t),type='series',t=0);
\[ y \relax (t ) = c_{1} t^{-i} \left (1+\left (-\frac {1}{5}-\frac {2 i}{5}\right ) t +\left (-\frac {1}{40}+\frac {3 i}{40}\right ) t^{2}+\left (\frac {3}{520}-\frac {7 i}{1560}\right ) t^{3}+\left (-\frac {1}{2496}+\frac {i}{12480}\right ) t^{4}+\left (\frac {9}{603200}+\frac {i}{361920}\right ) t^{5}+\mathrm {O}\left (t^{6}\right )\right )+c_{2} t^{i} \left (1+\left (-\frac {1}{5}+\frac {2 i}{5}\right ) t +\left (-\frac {1}{40}-\frac {3 i}{40}\right ) t^{2}+\left (\frac {3}{520}+\frac {7 i}{1560}\right ) t^{3}+\left (-\frac {1}{2496}-\frac {i}{12480}\right ) t^{4}+\left (\frac {9}{603200}-\frac {i}{361920}\right ) t^{5}+\mathrm {O}\left (t^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.013 (sec). Leaf size: 90
AsymptoticDSolveValue[t^2*y''[t]+t*y'[t]+(1+t)*y[t]==0,y[t],{t,0,5}]
\[ y(t)\to \left (\frac {1}{12480}+\frac {i}{2496}\right ) c_2 t^{-i} \left (i t^4-(8+16 i) t^3+(168+96 i) t^2-(1056-288 i) t+(480-2400 i)\right )-\left (\frac {1}{2496}+\frac {i}{12480}\right ) c_1 t^i \left (t^4-(16+8 i) t^3+(96+168 i) t^2+(288-1056 i) t-(2400-480 i)\right ) \]