What is the general solution to the system of differential equations given by dx/dt = 7x – y and dy/dt = 5x – 3y?

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General Solution for the System of Differential Equations

To find the general solution of the system:

  • dx/dt = 7x - y
  • dy/dt = 5x - 3y

We start by rewriting the system in matrix form:


	
d/dt X = AX
		
		where 
		X = [
		x  
		y 
	], 
		A = 
		[ 7  -1 ]
		[ 5  -3 ]
		]

To solve this, we need to find the eigenvalues and eigenvectors of matrix A.

Step 1: Finding Eigenvalues

The characteristic equation is obtained by solving:

det(A - 
			lambda*I) = 0
		
det(
		[ 7 - 
				lambda  -1 ]
		[ 5  -3 - 
				lambda ]
		) = 0

This expands to:

(7 - 
			lambda)(-3 - 
			lambda) - (-1)(5) = 0
		
			o 
			lambda^2 - 4lambda + 26 = 0

Step 2: Solving the Characteristic Equation

Using the quadratic formula:


			lambda = rac{-b \\pm 
			ext{sqrt}(b^2 - 4ac)}{2a} = rac{4 \\pm 	ext{sqrt}((-4)^2 - 4(1)(26))}{2(1)}

Calculating under the radical:


		 = rac{4 \\pm 	ext{sqrt}(-100)}{2} = 2 \\pm 5i

Step 3: Finding Eigenvectors

For each eigenvalue
2 + 5i and
2 - 5i, substitute back to find the corresponding eigenvectors:


		(A - (2 \\pm 5i)I)X = 0

By simplifying and solving these we get the eigenvectors. Let’s denote them by
v_1 and
v_2.

Step 4: Constructing the General Solution

The general solution will be:
X(t) = c_1 e^{(2 + 5i)t} v_1 + c_2 e^{(2 - 5i)t} v_2,
where
c_1 and
c_2 are constants determined by initial conditions.

Since
e^{(2 + 5i)t} = e^{2t}( ext{cos}(5t) + i ext{sin}(5t)), we can break it down into real-valued functions which makes it easier to interpret in real-world applications.

Conclusion

The general solution combines these components and ultimately represents a family of curves in the
xy plane, allowing for a broad analysis of behavior over time.

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