90.19.8 problem 8

Internal problem ID [25290]
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 : 8
Date solved : Sunday, October 12, 2025 at 05:55:38 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d^{2}}{d t^{2}}y_{1} \left (t \right )+2 y_{1} \left (t \right )&=-3 y_{2} \left (t \right )\\ \frac {d^{2}}{d t^{2}}y_{2} \left (t \right )+2 \frac {d}{d t}y_{2} \left (t \right )-9 y_{2} \left (t \right )&=6 y_{1} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} y_{1} \left (0\right )&=-1 \\ D\left (y_{1} \right )\left (0\right )&=-4 \\ y_{2} \left (0\right )&=1 \\ D\left (y_{2} \right )\left (0\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.046 (sec). Leaf size: 29

ode:=[diff(diff(y__1(t),t),t)+2*y__1(t) = -3*y__2(t), diff(diff(y__2(t),t),t)+2*diff(y__2(t),t)-9*y__2(t) = 6*y__1(t)]; 
ic:=[y__1(0) = -1, D(y__1)(0) = -4, y__2(0) = 1, D(y__2)(0) = 2]; 
dsolve([ode,op(ic)]);
 

\begin{align*} y_{1} \left (t \right ) &= 6-7 \,{\mathrm e}^{t}+3 \,{\mathrm e}^{t} t \\ y_{2} \left (t \right ) &= -4+5 \,{\mathrm e}^{t}-3 \,{\mathrm e}^{t} t \\ \end{align*}

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 30

ode={D[y1[t],{t,2}]+2*y1[t]==-3*y2[t], D[y2[t],{t,2}]+2*D[y2[t],t]-9*y2[t]==6*y1[t]}; 
ic={y1[0]==-1,Derivative[1][y1][0] ==-4,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 (3 t-7)+6\\ \text {y2}(t)&\to e^t (5-3 t)-4 \end{align*}

✓ Sympy. Time used: 0.158 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
y1 = Function("y1") 
y2 = Function("y2") 
ode=[Eq(2*y1(t) + 3*y2(t) + Derivative(y1(t), (t, 2)),0),Eq(-6*y1(t) - 9*y2(t) + 2*Derivative(y2(t), t) + Derivative(y2(t), (t, 2)),0)] 
ics = {y1(0): -1, Subs(Derivative(y1(t), t), t, 0): -4, y2(0): 1, Subs(Derivative(y2(t), t), t, 0): 2} 
dsolve(ode,func=[y1(t),y2(t)],ics=ics)
 

\[ \left [ y_{1}{\left (t \right )} = 3 t e^{t} - 7 e^{t} + 6, \ y_{2}{\left (t \right )} = - 3 t e^{t} + 5 e^{t} - 4\right ] \]