Question

Solve the given initial-value problem.
dx/dt​=−7x−y
dy/dt​=9x−y
x(1)=0,y(1)=1​

Answer 1

To solve the given initial-value problem, we can use the method of solving a system of linear differential equations.

Step-by-step explanation:

To solve the given initial-value problem:

dx/dt = -7x - y
dy/dt = 9x - y

with x(1) = 0 and y(1) = 1,

we can use the method of solving a system of linear differential equations.
First, let's differentiate the first equation with respect to t:

d²x/dt² = -7(dx/dt) - (dy/dt)

Now we can substitute dx/dt and dy/dt with the given equations:

d²x/dt² = -7(-7x - y) - (9x - y)

Simplifying, we get:

d²x/dt² = 49x + 7y - 9x + y

d²x/dt² = 40x + 8y

This is a second-order linear homogeneous differential equation that can be solved using various techniques such as the characteristic equation or Laplace transforms to find the general solution.