61.7.11 problem Problem 5(c)

Internal problem ID [15403]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 8.3 Systems of Linear Differential Equations (Variation of Parameters). Problems page 514
Problem number : Problem 5(c)
Date solved : Thursday, October 02, 2025 at 10:12:43 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=7 x \left (t \right )-4 y+10 \,{\mathrm e}^{t}\\ y^{\prime }&=3 x \left (t \right )+14 y+6 \,{\mathrm e}^{2 t} \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=1 \\ y \left (0\right )&=-1 \\ \end{align*}

✓ Maple. Time used: 0.195 (sec). Leaf size: 53

ode:=[diff(x(t),t) = 7*x(t)-4*y(t)+10*exp(t), diff(y(t),t) = 3*x(t)+14*y(t)+6*exp(2*t)]; 
ic:=[x(0) = 1, y(0) = -1]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= \frac {67 \,{\mathrm e}^{10 t}}{9}-\frac {14 \,{\mathrm e}^{11 t}}{3}-\frac {13 \,{\mathrm e}^{t}}{9}-\frac {{\mathrm e}^{2 t}}{3} \\ y \left (t \right ) &= -\frac {67 \,{\mathrm e}^{10 t}}{12}+\frac {14 \,{\mathrm e}^{11 t}}{3}+\frac {{\mathrm e}^{t}}{3}-\frac {5 \,{\mathrm e}^{2 t}}{12} \\ \end{align*}

✓ Mathematica. Time used: 0.043 (sec). Leaf size: 64

ode={D[x[t],t]==7*x[t]-4*y[t]+10*Exp[t],D[y[t],t]==3*x[t]+14*y[t]+6*Exp[2*t]}; 
ic={x[0]==1,y[0]==-1}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to -\frac {1}{9} e^t \left (3 e^t-67 e^{9 t}+42 e^{10 t}+13\right )\\ y(t)&\to \frac {1}{12} e^t \left (-5 e^t-67 e^{9 t}+56 e^{10 t}+4\right ) \end{align*}

✓ Sympy. Time used: 0.141 (sec). Leaf size: 63

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-7*x(t) + 4*y(t) - 10*exp(t) + Derivative(x(t), t),0),Eq(-3*x(t) - 14*y(t) - 6*exp(2*t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - C_{1} e^{11 t} - \frac {4 C_{2} e^{10 t}}{3} - \frac {e^{2 t}}{3} - \frac {13 e^{t}}{9}, \ y{\left (t \right )} = C_{1} e^{11 t} + C_{2} e^{10 t} - \frac {5 e^{2 t}}{12} + \frac {e^{t}}{3}\right ] \]