60.9.40 problem 1895

Internal problem ID [11819]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1895
Date solved : Sunday, March 30, 2025 at 09:15:59 PM
CAS classification : system_of_ODEs

\begin{align*} \operatorname {a11} \left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right )+\operatorname {b11} \left (\frac {d}{d t}x \left (t \right )\right )+\operatorname {c11} x \left (t \right )+\operatorname {a12} \left (\frac {d^{2}}{d t^{2}}y \left (t \right )\right )+\operatorname {b12} \left (\frac {d}{d t}y \left (t \right )\right )+\operatorname {c12} y \left (t \right )&=0\\ \operatorname {a21} \left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right )+\operatorname {b21} \left (\frac {d}{d t}x \left (t \right )\right )+\operatorname {c21} x \left (t \right )+\operatorname {a22} \left (\frac {d^{2}}{d t^{2}}y \left (t \right )\right )+\operatorname {b22} \left (\frac {d}{d t}y \left (t \right )\right )+\operatorname {c22} y \left (t \right )&=0 \end{align*}

✓ Maple. Time used: 0.240 (sec). Leaf size: 1186

ode:=[a11*diff(diff(x(t),t),t)+b11*diff(x(t),t)+c11*x(t)+a12*diff(diff(y(t),t),t)+b12*diff(y(t),t)+c12*y(t) = 0, a21*diff(diff(x(t),t),t)+b21*diff(x(t),t)+c21*x(t)+a22*diff(diff(y(t),t),t)+b22*diff(y(t),t)+c22*y(t) = 0]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= \moverset {4}{\munderset {\textit {\_a} =1}{\sum }}{\mathrm e}^{\operatorname {RootOf}\left (\left (\operatorname {a11} \operatorname {a22} -\operatorname {a12} \operatorname {a21} \right ) \textit {\_Z}^{4}+\left (\operatorname {a11} \operatorname {b22} -\operatorname {a12} \operatorname {b21} -\operatorname {a21} \operatorname {b12} +\operatorname {a22} \operatorname {b11} \right ) \textit {\_Z}^{3}+\left (\operatorname {a11} \operatorname {c22} -\operatorname {a12} \operatorname {c21} -\operatorname {a21} \operatorname {c12} +\operatorname {c11} \operatorname {a22} +\operatorname {b11} \operatorname {b22} -\operatorname {b12} \operatorname {b21} \right ) \textit {\_Z}^{2}+\left (\operatorname {b11} \operatorname {c22} -\operatorname {b12} \operatorname {c21} -\operatorname {b21} \operatorname {c12} +\operatorname {b22} \operatorname {c11} \right ) \textit {\_Z} +\operatorname {c11} \operatorname {c22} -\operatorname {c12} \operatorname {c21} , \operatorname {index} =\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}} \\ \text {Expression too large to display} \\ \end{align*}

✓ Mathematica. Time used: 0.152 (sec). Leaf size: 7517

ode={a11*D[x[t],{t,2}]+b11*D[x[t],t]+c11*x[t]+a12*D[y[t],{t,2}]+b12*D[y[t],t]+c12*y[t]==0,a21*D[x[t],{t,2}]+b21*D[x[t],t]+c21*x[t]+a22*D[y[t],{t,2}]+b22*D[y[t],t]+c22*y[t]==0}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

Too large to display

✗ Sympy

from sympy import * 
t = symbols("t") 
a11 = symbols("a11") 
a12 = symbols("a12") 
a21 = symbols("a21") 
a22 = symbols("a22") 
b11 = symbols("b11") 
b12 = symbols("b12") 
b21 = symbols("b21") 
b22 = symbols("b22") 
c11 = symbols("c11") 
c12 = symbols("c12") 
c21 = symbols("c21") 
c22 = symbols("c22") 
x = Function("x") 
y = Function("y") 
ode=[Eq(a11*Derivative(x(t), (t, 2)) + a12*Derivative(y(t), (t, 2)) + b11*Derivative(x(t), t) + b12*Derivative(y(t), t) + c11*x(t) + c12*y(t),0),Eq(a21*Derivative(x(t), (t, 2)) + a22*Derivative(y(t), (t, 2)) + b21*Derivative(x(t), t) + b22*Derivative(y(t), t) + c21*x(t) + c22*y(t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

Timed Out