Find the particular solution?

Question

Dy/dt = x dx/dt = 6x-8y x=1 and y=-1 when t=0

Answer 1

The Differential Equation are dy/dt = x and dx/dt = 6x - 8y

Let us Consider

dy/dt = x      ..................Equation(1)

Apply derivative on both sides

               ..............Equation(2)

Now let us consider dx/dt = 6x - 8y

This is in the form of second order linear differential Equation.

then general solution of differential equation is where r ₁ and r₂ are the real roots of the differential equation

Let us consider a auxiliary equation as r² - 6r + 8 = 0

r² - 4r - 2r + 8 = 0

r(r - 4) -2(r - 4) = 0

(r - 4)(r - 2) = 0

r = 4, 2

Solution of the differential Equation is .

        ...................Equation(3)

Differentiate with respect to t

            ..................Equation(4)

Given Initial values are x = 1 and y = -1 when t = 0

Substitute initial values (t = 0 and y = -1) in Equation(3)

     ..................Equation(5)

Substitute initial values(x = 1) in Equation(1) then

dy/dt = x

dy / dt = 1

Substitute initial values(t = 0 and dy / dt = 1) in Equation(4)

     ..................Equation(6)

Multiply Equation(5) by 2

     ..................Equation(7)

Subtract Equation(7) from Equation(6)

Substitute the above value in Equation(5)

Therefore the solution of Differential Equation is .