87.20.28 problem 29

Internal problem ID [23713]
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 : 29
Date solved : Thursday, October 02, 2025 at 09:44:31 PM
CAS classification : system_of_ODEs

\begin{align*} c_{1}^{\prime }\left (t \right )&=-\frac {k c_{1} \left (t \right )}{V_{1}}+\frac {k c_{2} \left (t \right )}{V_{1}}\\ c_{2}^{\prime }\left (t \right )&=\frac {k c_{1} \left (t \right )}{V_{2}}-\frac {k c_{2} \left (t \right )}{V_{2}} \end{align*}

✓ Maple. Time used: 0.053 (sec). Leaf size: 52

ode:=[diff(c__1(t),t) = -k/V__1*c__1(t)+k/V__1*c__2(t), diff(c__2(t),t) = k/V__2*c__1(t)-k/V__2*c__2(t)]; 
dsolve(ode);
 

\begin{align*} c_{1} \left (t \right ) &= c_1 +c_2 \,{\mathrm e}^{-\frac {k \left (V_{1} +V_{2} \right ) t}{V_{1} V_{2}}} \\ c_{2} \left (t \right ) &= -\frac {c_2 \,{\mathrm e}^{-\frac {k \left (V_{1} +V_{2} \right ) t}{V_{1} V_{2}}} V_{1} -V_{2} c_1}{V_{2}} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 108

ode={D[c1[t],t]==-k/v1*c1[t]+k/v1*c2[t],D[c2[t],t]==k/v2*c1[t]-k/v2*c2[t]}; 
ic={}; 
DSolve[{ode,ic},{c1[t],c2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {c1}(t)&\to \frac {\text {v2} e^{-\frac {k t (\text {v1}+\text {v2})}{\text {v1} \text {v2}}} \left (c_2 \left (e^{\frac {k t (\text {v1}+\text {v2})}{\text {v1} \text {v2}}}-1\right )+c_1\right )+c_1 \text {v1}}{\text {v1}+\text {v2}}\\ \text {c2}(t)&\to \frac {\text {v1} e^{-\frac {k t (\text {v1}+\text {v2})}{\text {v1} \text {v2}}} \left (c_1 \left (e^{\frac {k t (\text {v1}+\text {v2})}{\text {v1} \text {v2}}}-1\right )+c_2\right )+c_2 \text {v2}}{\text {v1}+\text {v2}} \end{align*}

✓ Sympy. Time used: 0.112 (sec). Leaf size: 41

from sympy import * 
t = symbols("t") 
k = symbols("k") 
v1 = symbols("v1") 
v2 = symbols("v2") 
c1 = Function("c1") 
c2 = Function("c2") 
ode=[Eq(k*c1(t)/v1 - k*c2(t)/v1 + Derivative(c1(t), t),0),Eq(-k*c1(t)/v2 + k*c2(t)/v2 + Derivative(c2(t), t),0)] 
ics = {} 
dsolve(ode,func=[c1(t),c2(t)],ics=ics)
 

\[ \left [ c_{1}{\left (t \right )} = C_{1} - \frac {C_{2} v_{2} e^{- \frac {t \left (k v_{1} + k v_{2}\right )}{v_{1} v_{2}}}}{v_{1}}, \ c_{2}{\left (t \right )} = C_{1} + C_{2} e^{- \frac {t \left (k v_{1} + k v_{2}\right )}{v_{1} v_{2}}}\right ] \]