Internal problem ID [4526]
Book: Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson
2018.
Section: Chapter 8, Series solutions of differential equations. Section 8.4. page 449
Problem number: 16.
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 }+{\mathrm e}^{t} y=0} \end {gather*} With initial conditions \begin {align*} [y \relax (0) = 1, y^{\prime }\relax (0) = -1] \end {align*}
With the expansion point for the power series method at \(t = 0\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 20
Order:=6; dsolve([diff(y(t),t$2)+t*diff(y(t),t)+exp(t)*y(t)=0,y(0) = 1, D(y)(0) = -1],y(t),type='series',t=0);
\[ y \relax (t ) = 1-t -\frac {1}{2} t^{2}+\frac {1}{6} t^{3}+\frac {1}{6} t^{4}+\frac {1}{120} t^{5}+\mathrm {O}\left (t^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 36
AsymptoticDSolveValue[{y''[t]+t*y'[t]+Exp[t]*y[t]==0,{y[0]==1,y'[0]==-1}},y[t],{t,0,5}]
\[ y(t)\to \frac {t^5}{120}+\frac {t^4}{6}+\frac {t^3}{6}-\frac {t^2}{2}-t+1 \]