60.9.3 problem 1858

Internal problem ID [11782]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1858
Date solved : Wednesday, March 05, 2025 at 03:06:53 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.063 (sec). Leaf size: 63

ode:=[diff(x(t),t) = a*y(t), diff(y(t),t) = b*x(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{\sqrt {a}\, \sqrt {b}\, t}+c_{2} {\mathrm e}^{-\sqrt {a}\, \sqrt {b}\, t} \\ y \left (t \right ) &= \frac {\sqrt {b}\, \left (c_{1} {\mathrm e}^{\sqrt {a}\, \sqrt {b}\, t}-c_{2} {\mathrm e}^{-\sqrt {a}\, \sqrt {b}\, t}\right )}{\sqrt {a}} \\ \end{align*}

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 158

ode={D[x[t],t]==a*y[t],D[y[t],t]==b*x[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {e^{-\sqrt {a} \sqrt {b} t} \left (\sqrt {b} c_1 \left (e^{2 \sqrt {a} \sqrt {b} t}+1\right )+\sqrt {a} c_2 \left (e^{2 \sqrt {a} \sqrt {b} t}-1\right )\right )}{2 \sqrt {b}} \\ y(t)\to \frac {e^{-\sqrt {a} \sqrt {b} t} \left (\sqrt {b} c_1 \left (e^{2 \sqrt {a} \sqrt {b} t}-1\right )+\sqrt {a} c_2 \left (e^{2 \sqrt {a} \sqrt {b} t}+1\right )\right )}{2 \sqrt {a}} \\ \end{align*}

✓ Sympy. Time used: 0.169 (sec). Leaf size: 68

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

\[ \left [ x{\left (t \right )} = - \frac {C_{1} a e^{- t \sqrt {a b}}}{\sqrt {a b}} + \frac {C_{2} a e^{t \sqrt {a b}}}{\sqrt {a b}}, \ y{\left (t \right )} = C_{1} e^{- t \sqrt {a b}} + C_{2} e^{t \sqrt {a b}}\right ] \]