14.12.17 problem 17

Internal problem ID [2627]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.8. Series solutions. Excercises page 197
Problem number : 17
Date solved : Monday, January 27, 2025 at 06:04:49 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+y^{\prime }+{\mathrm e}^{-t} y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=3\\ y^{\prime }\left (0\right )&=5 \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 20

Order:=6; 
dsolve([diff(y(t),t$2)+diff(y(t),t)+exp(-t)*y(t)=0,y(0) = 3, D(y)(0) = 5],y(t),type='series',t=0);
 

\[ y = 3+5 t -4 t^{2}+t^{3}+\frac {3}{8} t^{4}-\frac {17}{40} t^{5}+\operatorname {O}\left (t^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 30

AsymptoticDSolveValue[{D[y[t],{t,2}]+D[y[t],t]+Exp[-t]*y[t]==0,{y[0]==3,Derivative[1][y][0] ==5}},y[t],{t,0,"6"-1}]
 

\[ y(t)\to -\frac {17 t^5}{40}+\frac {3 t^4}{8}+t^3-4 t^2+5 t+3 \]