87.20.29 problem 31

Internal problem ID [23714]
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 : 31
Date solved : Thursday, October 02, 2025 at 09:44:31 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=a \left (b -x\right )-c f y \left (t \right )\\ y^{\prime }\left (t \right )&=d \left (x-y \left (t \right )\right )-c f y \left (t \right )-a y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=b \\ y \left (0\right )&=\frac {d b}{a +d} \\ \end{align*}

✓ Maple. Time used: 0.083 (sec). Leaf size: 237

ode:=[diff(x(t),t) = a*(b-x(t))-c*f*y(t), diff(y(t),t) = d*(x(t)-y(t))-c*f*y(t)-a*y(t)]; 
ic:=[x(0) = b, y(0) = d*b/(a+d)]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= \frac {{\mathrm e}^{\left (-c f -a \right ) t} c f d b}{a c f +c d f +a^{2}+a d}+\frac {a b \left (c f +a +d \right )}{\left (a +d \right ) \left (c f +a \right )} \\ y \left (t \right ) &= \frac {\frac {{\mathrm e}^{-\left (c f +a \right ) t} c^{3} f^{3} d b a}{a c f +c d f +a^{2}+a d}+\frac {{\mathrm e}^{-\left (c f +a \right ) t} c^{3} f^{3} d^{2} b}{a c f +c d f +a^{2}+a d}+\frac {{\mathrm e}^{-\left (c f +a \right ) t} c^{2} f^{2} d b \,a^{2}}{a c f +c d f +a^{2}+a d}+\frac {{\mathrm e}^{-\left (c f +a \right ) t} c^{2} f^{2} d^{2} b a}{a c f +c d f +a^{2}+a d}+a b c d f}{\left (a +d \right ) \left (c f +a \right ) c f} \\ \end{align*}

✓ Mathematica. Time used: 0.061 (sec). Leaf size: 77

ode={D[x[t],t]==a*(b-x[t])-c*f*y[t],D[y[t],t]==d*(x[t]-y[t])-c*f*y[t]-a*y[t] }; 
ic={x[0]==b,y[0]==d*b/(a+d)}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {b \left (a^2+c d f e^{-(t (a+c f))}+a (c f+d)\right )}{(a+d) (a+c f)}\\ y(t)&\to \frac {b d \left (c f e^{-(t (a+c f))}+a\right )}{(a+d) (a+c f)} \end{align*}

✓ Sympy. Time used: 0.425 (sec). Leaf size: 371

from sympy import * 
t = symbols("t") 
b = symbols("b") 
d = symbols("d") 
a = symbols("a") 
c = symbols("c") 
f = symbols("f") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-a*(b - x(t)) + c*f*y(t) + Derivative(x(t), t),0),Eq(a*y(t) + c*f*y(t) - d*(x(t) - y(t)) + Derivative(y(t), t),0)] 
ics = {x(0): b, y(0): b*d/(a + d)} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \text {Solution too large to show} \]