Show that the origin is the only equilibrium point of the system?
X=ax+by
Y=cx+dy
Wherea,b,c and d are constants, with ad-bc not aqual to 0(zero)
Please show me how to do this problem
Please show me the work
To show that the origin (0, 0) is the only equilibrium point of the system, we need to find values of x and y for which both equations are satisfied simultaneously.
Let's consider the first equation: X = ax + by
At an equilibrium point, the values of X and Y should remain unchanged, which means X = 0. Therefore, we have:
0 = ax + by --(1)
Similarly, let's consider the second equation: Y = cx + dy
At an equilibrium point, the values of X and Y should remain unchanged, which means Y = 0. Therefore, we have:
0 = cx + dy --(2)
Now, we will solve these two equations simultaneously to find the values of x and y.
Multiplying equation (1) by d and equation (2) by b, we get:
0 = adx + bdy --(3)
0 = bcx + bdy --(4)
Subtracting equation (4) from equation (3), we obtain:
0 = (ad - bc)x
Since ad - bc is not equal to 0 (as given in the problem), we can divide both sides by (ad - bc):
x = 0
Substituting this result into equation (1), we find:
0 = 0 + by
0 = by
This implies that y = 0.
Hence, the only equilibrium point of the system is (x, y) = (0, 0), which corresponds to the origin.
To show that the origin is the only equilibrium point of the system, we need to demonstrate that the system reaches a steady state at the origin (0, 0) regardless of the initial conditions. In other words, we want to prove that when x = 0 and y = 0, the values of x and y remain at 0 as time progresses.
Let's start by setting up the differential equations representing the given system:
dX/dt = ax + by
dY/dt = cx + dy
To find the equilibrium points, we set dX/dt and dY/dt equal to 0:
ax + by = 0 ---- (1)
cx + dy = 0 ---- (2)
Now, we solve this system of equations to find the values of x and y that satisfy both equations.
First, we'll solve equation (1) for x:
x = -(by) / a
Next, we substitute this expression for x into equation (2):
c(-(by) / a) + dy = 0
-cby/a + dy = 0
(d - cb/a)y = 0
Since ad - bc ≠ 0, we know that cb/a ≠ d. Therefore, the only solution for y is y = 0.
Now, substituting y = 0 back into equation (1), we have:
ax + b(0) = 0
ax = 0
Since we assume a ≠ 0, the only solution for x is x = 0.
Therefore, the only values that satisfy both equations (1) and (2) are x = 0 and y = 0, which means that the origin (0, 0) is the only equilibrium point of the system.
To summarize:
- We set dX/dt and dY/dt to 0 and solved the resulting system of equations.
- Through the solving process, we found that the only values that satisfy both equations are x = 0 and y = 0, indicating that the origin is the only equilibrium point of the system.