Find the general solution of the given system. dx / dt = 7x − y dy / dt = 5x + 3y.

We will use the concept of the parametric form of the expression in order to find the general form.

Answer: General solution of the given system dx / dt = 7x − y dy / dt = 5x + 3y is x(t) = e^5t (A cost + B sint) and y(t) = e^5t [(2A + B) cost + (2B + A) sint]

Let us see how we will use the concept of the parametric form of the expression in order to find the general form.

Explanation:

The equations  in the parametric form are given as,

dx / dt = 7x − y

dy / dt = 5x + 3y

⇒ x' = 7x − y ------ (1)

y' = 5x + 3y ------ (2)

Differentiate equation (1) w.r.t. t

x'' = 7 x' - y ' 

= 7x' - 5x - 3y (From equation (2))

= 7x' - 5x - 3(7x - x') (From equation (1))

= 7x' - 5x - 21x + 3x'

= 10x' - 26x

⇒ x'' - 10x' + 26x = 0

The auxillary equation of the differential equaltion x'' - 10x' + 26x = 0 is r² - 10r + 26 = 0. Now, find the roots of the equation r² - 10r + 26 = 0 using the quadratic formula.

r = [10 ± √ (100 - 104)]/2

= [10 ± √ -4]/2

= (10 ± 2i)/2

= 5 ± i

The general solution of the second-order equation for x is 

x(t) = e^5t (A cost + B sint)

x'(t) = 5e^5t (A cost + B sint) + e^5t (-A sint + B cost)

From equation (1), we have

y(t) = 7x - x'

= 7[e^5t (A cost + B sint)] - [5e^5t (A cost + B sint) + e^5t (-A sint + B cost)]

= e^5t [cost  (7A - 5A + B) + sint (7B - 5B + A)]

= e^5t [(2A + B) cost + (2B + A) sint]

Hence, the general solution is x(t) = e^5t (A cost + B sint) and y(t) = e^5t [(2A + B) cost + (2B + A) sint]