14.12.16 problem 16

Internal problem ID [2626]
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 : 16
Date solved : Monday, January 27, 2025 at 06:04:48 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 )&=0\\ y^{\prime }\left (0\right )&=-1 \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 16

Order:=6; 
dsolve([diff(y(t),t$2)+diff(y(t),t)+exp(t)*y(t)=0,y(0) = 0, D(y)(0) = -1],y(t),type='series',t=0);
 

\[ y = -t +\frac {1}{2} t^{2}+\frac {1}{24} t^{4}-\frac {1}{120} t^{5}+\operatorname {O}\left (t^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 28

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

\[ y(t)\to -\frac {t^5}{120}+\frac {t^4}{24}+\frac {t^2}{2}-t \]