54.9.37 problem 1892

Internal problem ID [13115]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1892
Date solved : Sunday, October 12, 2025 at 02:28:24 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d^{2}}{d t^{2}}x \left (t \right )-a \left (\frac {d}{d t}y \left (t \right )\right )+b x \left (t \right )&=0\\ \frac {d^{2}}{d t^{2}}y \left (t \right )+a \left (\frac {d}{d t}x \left (t \right )\right )+b y \left (t \right )&=0 \end{align*}

✓ Maple. Time used: 0.196 (sec). Leaf size: 867

ode:=[diff(diff(x(t),t),t)-a*diff(y(t),t)+b*x(t) = 0, diff(diff(y(t),t),t)+a*diff(x(t),t)+b*y(t) = 0]; 
dsolve(ode);
 

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 0.138 (sec). Leaf size: 3522

ode={D[x[t],{t,2}]-a*D[y[t],t]+b*x[t]==0,D[y[t],{t,2}]+a*D[x[t],t]+b*y[t]==0}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

Too large to display

✓ Sympy. Time used: 0.873 (sec). Leaf size: 450

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

\[ \left [ x{\left (t \right )} = - \frac {2 C_{1} e^{- \frac {\sqrt {2} t \sqrt {- a^{2} + a \sqrt {a^{2} + 4 b} - 2 b}}{2}}}{a - \sqrt {a^{2} + 4 b}} - \frac {2 C_{2} e^{\frac {\sqrt {2} t \sqrt {- a^{2} + a \sqrt {a^{2} + 4 b} - 2 b}}{2}}}{a - \sqrt {a^{2} + 4 b}} - \frac {2 C_{3} e^{- \frac {\sqrt {2} t \sqrt {- a^{2} - a \sqrt {a^{2} + 4 b} - 2 b}}{2}}}{a + \sqrt {a^{2} + 4 b}} - \frac {2 C_{4} e^{\frac {\sqrt {2} t \sqrt {- a^{2} - a \sqrt {a^{2} + 4 b} - 2 b}}{2}}}{a + \sqrt {a^{2} + 4 b}}, \ y{\left (t \right )} = - \frac {\sqrt {2} C_{1} e^{- \frac {\sqrt {2} t \sqrt {- a^{2} + a \sqrt {a^{2} + 4 b} - 2 b}}{2}}}{\sqrt {- a^{2} + a \sqrt {a^{2} + 4 b} - 2 b}} + \frac {\sqrt {2} C_{2} e^{\frac {\sqrt {2} t \sqrt {- a^{2} + a \sqrt {a^{2} + 4 b} - 2 b}}{2}}}{\sqrt {- a^{2} + a \sqrt {a^{2} + 4 b} - 2 b}} - \frac {\sqrt {2} C_{3} e^{- \frac {\sqrt {2} t \sqrt {- a^{2} - a \sqrt {a^{2} + 4 b} - 2 b}}{2}}}{\sqrt {- a^{2} - a \sqrt {a^{2} + 4 b} - 2 b}} + \frac {\sqrt {2} C_{4} e^{\frac {\sqrt {2} t \sqrt {- a^{2} - a \sqrt {a^{2} + 4 b} - 2 b}}{2}}}{\sqrt {- a^{2} - a \sqrt {a^{2} + 4 b} - 2 b}}\right ] \]