87.20.9 problem 10

Internal problem ID [23694]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 3. Linear Systems. Exercise at page 161
Problem number : 10
Date solved : Thursday, October 02, 2025 at 09:44:20 PM
CAS classification : system_of_ODEs

\begin{align*} N_{1}^{\prime }\left (t \right )&=4 N_{1} \left (t \right )-6 N_{2} \left (t \right )\\ N_{2}^{\prime }\left (t \right )&=8 N_{1} \left (t \right )-10 N_{2} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} N_{1} \left (0\right )&=100000 \\ N_{2} \left (0\right )&=1000 \\ \end{align*}

✓ Maple. Time used: 0.050 (sec). Leaf size: 33

ode:=[diff(N__1(t),t) = 4*N__1(t)-6*N__2(t), diff(N__2(t),t) = 8*N__1(t)-10*N__2(t)]; 
ic:=[N__1(0) = 100000, N__2(0) = 1000]; 
dsolve([ode,op(ic)]);
 

\begin{align*} N_{1} \left (t \right ) &= -297000 \,{\mathrm e}^{-4 t}+397000 \,{\mathrm e}^{-2 t} \\ N_{2} \left (t \right ) &= -396000 \,{\mathrm e}^{-4 t}+397000 \,{\mathrm e}^{-2 t} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 40

ode={D[N1[t],t]==4*N1[t]-6*N2[t],D[N2[t],t]==8*N1[t]-10*N2[t]}; 
ic={N1[0]==10^5,N2[0]==10^3}; 
DSolve[{ode,ic},{N1[t],N2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {N1}(t)&\to 1000 e^{-4 t} \left (397 e^{2 t}-297\right )\\ \text {N2}(t)&\to 1000 e^{-4 t} \left (397 e^{2 t}-396\right ) \end{align*}

✓ Sympy. Time used: 0.078 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
N1 = Function("N1") 
N2 = Function("N2") 
ode=[Eq(-4*N1(t) + 6*N2(t) + Derivative(N1(t), t),0),Eq(-8*N1(t) + 10*N2(t) + Derivative(N2(t), t),0)] 
ics = {N1(0): 100000, N2(0): 1000} 
dsolve(ode,func=[N1(t),N2(t)],ics=ics)
 

\[ \left [ N_{1}{\left (t \right )} = 397000 e^{- 2 t} - 297000 e^{- 4 t}, \ N_{2}{\left (t \right )} = 397000 e^{- 2 t} - 396000 e^{- 4 t}\right ] \]