90.19.5 problem 5

Internal problem ID [25287]
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 : 5
Date solved : Sunday, October 12, 2025 at 05:55:37 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )+4 y_{1} \left (t \right )&=10 y_{2} \left (t \right )\\ \frac {d^{2}}{d t^{2}}y_{2} \left (t \right )-6 \frac {d}{d t}y_{2} \left (t \right )+23 y_{2} \left (t \right )&=9 y_{1} \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.046 (sec). Leaf size: 41

ode:=[diff(y__1(t),t)+4*y__1(t) = 10*y__2(t), diff(diff(y__2(t),t),t)-6*diff(y__2(t),t)+23*y__2(t) = 9*y__1(t)]; 
ic:=[y__1(0) = 0, y__2(0) = 2, D(y__2)(0) = 2]; 
dsolve([ode,op(ic)]);
 

\begin{align*} y_{1} \left (t \right ) &= -20 \,{\mathrm e}^{2 t}-20 \,{\mathrm e}^{-t}+40 \,{\mathrm e}^{t} \\ y_{2} \left (t \right ) &= -12 \,{\mathrm e}^{2 t}-6 \,{\mathrm e}^{-t}+20 \,{\mathrm e}^{t} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 49

ode={D[y1[t],t]+4*y1[t]==10*y2[t], D[y2[t],{t,2}]-6*D[y2[t],t]+23*y2[t]==9*y1[t]}; 
ic={y1[0]==0,y2[0]==2,Derivative[1][y2][0] ==2}; 
DSolve[{ode,ic},{y1[t],y2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)&\to -20 e^{-t} \left (-2 e^{2 t}+e^{3 t}+1\right )\\ \text {y2}(t)&\to -6 e^{-t}+20 e^t-12 e^{2 t} \end{align*}

✓ Sympy. Time used: 0.111 (sec). Leaf size: 37

from sympy import * 
t = symbols("t") 
y1 = Function("y1") 
y2 = Function("y2") 
ode=[Eq(4*y1(t) - 10*y2(t) + Derivative(y1(t), t),0),Eq(-9*y1(t) + 23*y2(t) - 6*Derivative(y2(t), t) + Derivative(y2(t), (t, 2)),0)] 
ics = {y1(0): 0, y2(0): 2, Subs(Derivative(y2(t), t), t, 0): 2} 
dsolve(ode,func=[y1(t),y2(t)],ics=ics)
 

\[ \left [ y_{1}{\left (t \right )} = - 20 e^{2 t} + 40 e^{t} - 20 e^{- t}, \ y_{2}{\left (t \right )} = - 12 e^{2 t} + 20 e^{t} - 6 e^{- t}\right ] \]