Internal problem ID [1780]
Book: Differential equations and their applications, 3rd ed., M. Braun
Section: Section 2.8, Series solutions. Page 195
Problem number: 15.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+t y^{\prime }+y \,{\mathrm e}^{t}=0} \end {gather*} With initial conditions \begin {align*} [y \relax (0) = 1, y^{\prime }\relax (0) = 0] \end {align*}
With the expansion point for the power series method at \(t = 0\).
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 18
Order:=6; dsolve([diff(y(t),t$2)+t*diff(y(t),t)+exp(t)*y(t)=0,y(0) = 1, D(y)(0) = 0],y(t),type='series',t=0);
\[ y \relax (t ) = 1-\frac {1}{2} t^{2}-\frac {1}{6} t^{3}+\frac {1}{12} t^{4}+\frac {1}{20} t^{5}+\mathrm {O}\left (t^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 33
AsymptoticDSolveValue[{y''[t]+t*y'[t]+Exp[t]*y[t]==0,{y[0]==1,y'[0]==0}},y[t],{t,0,5}]
\[ y(t)\to \frac {t^5}{20}+\frac {t^4}{12}-\frac {t^3}{6}-\frac {t^2}{2}+1 \]