dx/dt=5x-2y,dy/dt=4x-y
To solve this system of differential equations, we can use a method called "elimination." The goal is to eliminate one of the variables, either x or y, by manipulating the equations so that one equation only contains derivatives of the same variable.
Let's start by manipulating the first equation, dx/dt = 5x - 2y:
1. Multiply both sides of the equation by 4: 4(dx/dt) = 4(5x - 2y)
This gives us: 4dx/dt = 20x - 8y
Now, let's manipulate the second equation, dy/dt = 4x - y:
2. Multiply both sides of the equation by 2: 2(dy/dt) = 2(4x - y)
This gives us: 2dy/dt = 8x - 2y
Now we have two equations: 4dx/dt = 20x - 8y and 2dy/dt = 8x - 2y.
Next, let's multiply the second equation by 2 to make the coefficients of dy/dt and dx/dt the same:
3. Multiply both sides of the second equation by 2: 4(dy/dt) = 4(8x - 2y)
This gives us: 4dy/dt = 32x - 8y
Now, we have three equations: 4dx/dt = 20x - 8y, 4dy/dt = 32x - 8y, and 4dy/dt = 32x - 8y.
To eliminate y, we can subtract the second equation from the third equation:
4. 4dy/dt - 4dy/dt = 32x - 8y - (32x - 8y)
This simplifies to: 0 = 0
Since 0 = 0 is a true statement, it means that the third equation is redundant and does not provide any additional information.
Therefore, the system of differential equations can be simplified to:
4dx/dt = 20x - 8y
4dy/dt = 32x - 8y
Now we have a system of two first-order differential equations, which we can solve using various methods such as separation of variables, substitution, or matrix methods.