7.12.18 problem 17

Internal problem ID [2430]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.8, Series solutions. Page 195
Problem number : 17
Date solved : Tuesday, September 30, 2025 at 05:35:44 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*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 20

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

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

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 30

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

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

✓ Sympy. Time used: 0.371 (sec). Leaf size: 80

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(y(t)*exp(-t) + Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 3, Subs(Derivative(y(t), t), t, 0): 5} 
dsolve(ode,func=y(t),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (t \right )} = C_{2} \left (- \frac {t^{4} e^{- t}}{24} + \frac {t^{4} e^{- 2 t}}{24} + \frac {t^{3} e^{- t}}{6} - \frac {t^{2} e^{- t}}{2} + 1\right ) + C_{1} t \left (- \frac {t^{3}}{24} + \frac {t^{3} e^{- t}}{12} + \frac {t^{2}}{6} - \frac {t^{2} e^{- t}}{6} - \frac {t}{2} + 1\right ) + O\left (t^{6}\right ) \]