66.11.6 problem 8

Internal problem ID [16126]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 3. Linear Systems. Exercises section 3.4 page 310
Problem number : 8
Date solved : Thursday, October 02, 2025 at 10:42:44 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=x \left (t \right )+4 y\\ y^{\prime }&=-3 x \left (t \right )+2 y \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.169 (sec). Leaf size: 60

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{\frac {3 t}{2}} \left (-\frac {9 \sqrt {47}\, \sin \left (\frac {\sqrt {47}\, t}{2}\right )}{47}+\cos \left (\frac {\sqrt {47}\, t}{2}\right )\right ) \\ y \left (t \right ) &= -\frac {{\mathrm e}^{\frac {3 t}{2}} \left (\frac {56 \sqrt {47}\, \sin \left (\frac {\sqrt {47}\, t}{2}\right )}{47}+8 \cos \left (\frac {\sqrt {47}\, t}{2}\right )\right )}{8} \\ \end{align*}

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 94

ode={D[x[t],t]==1*x[t]+4*y[t],D[y[t],t]==-3*x[t]+2*y[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}{47} e^{3 t/2} \left (47 \cos \left (\frac {\sqrt {47} t}{2}\right )-9 \sqrt {47} \sin \left (\frac {\sqrt {47} t}{2}\right )\right )\\ y(t)&\to -\frac {1}{47} e^{3 t/2} \left (7 \sqrt {47} \sin \left (\frac {\sqrt {47} t}{2}\right )+47 \cos \left (\frac {\sqrt {47} t}{2}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.119 (sec). Leaf size: 99

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

\[ \left [ x{\left (t \right )} = \left (\frac {C_{1}}{6} + \frac {\sqrt {47} C_{2}}{6}\right ) e^{\frac {3 t}{2}} \cos {\left (\frac {\sqrt {47} t}{2} \right )} + \left (\frac {\sqrt {47} C_{1}}{6} - \frac {C_{2}}{6}\right ) e^{\frac {3 t}{2}} \sin {\left (\frac {\sqrt {47} t}{2} \right )}, \ y{\left (t \right )} = C_{1} e^{\frac {3 t}{2}} \cos {\left (\frac {\sqrt {47} t}{2} \right )} - C_{2} e^{\frac {3 t}{2}} \sin {\left (\frac {\sqrt {47} t}{2} \right )}\right ] \]