57.23.4 problem 6

Internal problem ID [14525]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 4, Linear Systems. Exercises page 244
Problem number : 6
Date solved : Thursday, October 02, 2025 at 09:38:09 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=3 x+2 y \left (t \right )+3\\ y^{\prime }\left (t \right )&=7 x+5 y \left (t \right )+2 t \end{align*}

✓ Maple. Time used: 0.129 (sec). Leaf size: 90

ode:=[diff(x(t),t) = 3*x(t)+2*y(t)+3, diff(y(t),t) = 7*x(t)+5*y(t)+2*t]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{\left (4+\sqrt {15}\right ) t} c_2 +{\mathrm e}^{-\left (-4+\sqrt {15}\right ) t} c_1 +4 t +17 \\ y \left (t \right ) &= \frac {{\mathrm e}^{\left (4+\sqrt {15}\right ) t} c_2 \sqrt {15}}{2}-\frac {{\mathrm e}^{-\left (-4+\sqrt {15}\right ) t} c_1 \sqrt {15}}{2}+\frac {{\mathrm e}^{\left (4+\sqrt {15}\right ) t} c_2}{2}+\frac {{\mathrm e}^{-\left (-4+\sqrt {15}\right ) t} c_1}{2}-6 t -25 \\ \end{align*}

✓ Mathematica. Time used: 0.698 (sec). Leaf size: 178

ode={D[x[t],t]==3*x[t]+2*y[t],D[y[t],t]==7*x[t]+5*y[t]+2*t}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{30} e^{-\left (\left (\sqrt {15}-4\right ) t\right )} \left (120 e^{\left (\sqrt {15}-4\right ) t} (t+8)+\left (2 \sqrt {15} c_2-\left (\sqrt {15}-15\right ) c_1\right ) e^{2 \sqrt {15} t}+\left (15+\sqrt {15}\right ) c_1-2 \sqrt {15} c_2\right )\\ y(t)&\to \frac {1}{30} e^{-\left (\left (\sqrt {15}-4\right ) t\right )} \left (-60 e^{\left (\sqrt {15}-4\right ) t} (3 t+23)+\left (7 \sqrt {15} c_1+\left (15+\sqrt {15}\right ) c_2\right ) e^{2 \sqrt {15} t}-7 \sqrt {15} c_1-\left (\sqrt {15}-15\right ) c_2\right ) \end{align*}

✓ Sympy. Time used: 0.366 (sec). Leaf size: 78

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-3*x(t) - 2*y(t) + Derivative(x(t), t) - 3,0),Eq(-2*t - 7*x(t) - 5*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \frac {C_{1} \left (1 + \sqrt {15}\right ) e^{t \left (4 - \sqrt {15}\right )}}{7} - \frac {C_{2} \left (1 - \sqrt {15}\right ) e^{t \left (\sqrt {15} + 4\right )}}{7} + 4 t + 17, \ y{\left (t \right )} = C_{1} e^{t \left (4 - \sqrt {15}\right )} + C_{2} e^{t \left (\sqrt {15} + 4\right )} - 6 t - 25\right ] \]