54.9.6 problem 1861

Internal problem ID [13084]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1861
Date solved : Wednesday, October 01, 2025 at 03:34:21 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.167 (sec). Leaf size: 134

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

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{\frac {\left (i a \beta -i b \alpha +a \alpha +b \beta \right ) t}{a^{2}+b^{2}}}+c_2 \,{\mathrm e}^{-\frac {\left (i a \beta -i b \alpha -a \alpha -b \beta \right ) t}{a^{2}+b^{2}}} \\ y \left (t \right ) &= i \left (c_1 \,{\mathrm e}^{\frac {\left (i a \beta -i b \alpha +a \alpha +b \beta \right ) t}{a^{2}+b^{2}}}-c_2 \,{\mathrm e}^{-\frac {\left (i a \beta -i b \alpha -a \alpha -b \beta \right ) t}{a^{2}+b^{2}}}\right ) \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 145

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

\begin{align*} x(t)&\to e^{\frac {t (a \alpha +b \beta )}{a^2+b^2}} \left (c_1 \cos \left (\frac {t (a \beta -\alpha b)}{a^2+b^2}\right )+c_2 \sin \left (\frac {t (a \beta -\alpha b)}{a^2+b^2}\right )\right )\\ y(t)&\to e^{\frac {t (a \alpha +b \beta )}{a^2+b^2}} \left (c_2 \cos \left (\frac {t (a \beta -\alpha b)}{a^2+b^2}\right )-c_1 \sin \left (\frac {t (a \beta -\alpha b)}{a^2+b^2}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.377 (sec). Leaf size: 122

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

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