90.19.11 problem 14

Internal problem ID [25293]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 4. Linear Constant Coefficient Differential Equations. Exercises at page 309
Problem number : 14
Date solved : Sunday, October 12, 2025 at 05:55:38 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )-2 y_{1} \left (t \right )&=-y_{2} \left (t \right )\\ \frac {d^{2}}{d t^{2}}y_{2} \left (t \right )-\frac {d}{d t}y_{2} \left (t \right )+y_{2} \left (t \right )&=y_{1} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} y_{1} \left (0\right )&=0 \\ y_{2} \left (0\right )&=-1 \\ D\left (y_{2} \right )\left (0\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.043 (sec). Leaf size: 30

ode:=[diff(y__1(t),t)-2*y__1(t) = -y__2(t), diff(diff(y__2(t),t),t)-diff(y__2(t),t)+y__2(t) = y__1(t)]; 
ic:=[y__1(0) = 0, y__2(0) = -1, D(y__2)(0) = 2]; 
dsolve([ode,op(ic)]);
 

\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{t} \left (-t^{2}+t \right ) \\ y_{2} \left (t \right ) &= {\mathrm e}^{t} \left (-t^{2}+3 t -1\right ) \\ \end{align*}

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

ode={D[y1[t],{t,1}]-2*y1[t]==-y2[t], D[y2[t],{t,2}]-D[y2[t],t]+y2[t]==y1[t]}; 
ic={y1[0]==0,y2[0]==-1,Derivative[1][y2][0] ==2}; 
DSolve[{ode,ic},{y1[t],y2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)&\to -e^t (t-1) t\\ \text {y2}(t)&\to -e^t \left (t^2-3 t+1\right ) \end{align*}

✓ Sympy. Time used: 0.109 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
y1 = Function("y1") 
y2 = Function("y2") 
ode=[Eq(-2*y1(t) + y2(t) + Derivative(y1(t), t),0),Eq(-y1(t) + y2(t) - Derivative(y2(t), t) + Derivative(y2(t), (t, 2)),0)] 
ics = {y1(0): 2, y2(0): -1, Subs(Derivative(y2(t), t), t, 0): 2} 
dsolve(ode,func=[y1(t),y2(t)],ics=ics)
 

\[ \left [ y_{1}{\left (t \right )} = 3 t e^{t} + 2 e^{t}, \ y_{2}{\left (t \right )} = 3 t e^{t} - e^{t}\right ] \]