Question

Find the solution of the system

x' = -4y, y' = -6x,
where primes indicate derivatives with respect to t, that satisfies the initial condition x(0) = 2, y(0) = -2.
x =___
y =____

Answer 1

To solve the system of differential equations x' = -4y and y' = -6x with initial conditions x(0) = 2 and y(0) = -2, first use the method of separation of variables. Then, substitute the initial conditions to find the values of the constants. The solution to the system is x = -4yt + 2 and y = -3x^2 - 2.

Step-by-step explanation:

To solve the system of differential equations x' = -4y and y' = -6x with initial conditions x(0) = 2 and y(0) = -2, we can use the method of separation of variables. First, let's solve the equation x' = -4y. We can rewrite this as dx/dt = -4y. Separating variables, we have dx = -4y dt. Integrating both sides, we get x = -4yt + C, where C is the constant of integration.

Next, let's solve the equation y' = -6x. We can rewrite this as dy/dt = -6x. Separating variables, we have dy = -6x dt. Integrating both sides, we get y = -3x^2 + D, where D is the constant of integration.

Using the initial conditions x(0) = 2 and y(0) = -2, we can substitute these values into the equations x = -4yt + C and y = -3x^2 + D. Solving for the constants C and D, we find C = 2 and D = -2. Therefore, the solution to the system of equations is x = -4yt + 2 and y = -3x^2 - 2.