Answer:
Explanation:
To solve the system of differential equations:
dx/dt = 3x - 18y ...(1)
dy/dt = 2x - 9y ...(2)
Step 1: Solve for x in equation (2):
2x = dy/dt + 9y
x = (1/2) (dy/dt + 9y)
Step 2: Substitute the expression for x from step 1 into equation (1):
dx/dt = 3x - 18y
dx/dt = 3[(1/2) (dy/dt + 9y)] - 18y
dx/dt = (3/2) dy/dt + (9/2) y - 18y
dx/dt = (3/2) dy/dt - (9/2) y
Step 3: Rewrite the equation from step 2 in terms of y and its derivative:
dx/dt + (9/2) y = (3/2) dy/dt
Step 4: Solve for y(t) using the integrating factor method:
Multiplying both sides by exp(9t/2), we get:
exp(9t/2) dx/dt + (9/2) exp(9t/2) y = (3/2) exp(9t/2) dy/dt
Applying the product rule on the left-hand side:
d/dt (exp(9t/2) x) = (3/2) d/dt (exp(9t/2) y)
Integrating both sides with respect to t:
exp(9t/2) x = (3/2) exp(9t/2) y + C
where C is the constant of integration.
Step 5: Solve for x(t) using the expression for y(t) from step 4:
x = (3/2) y + C exp(-9t/2)
where C is the constant of integration.
Therefore, the general solution to the system of differential equations is:
x = (3/2) y + C exp(-9t/2)
y = exp(9t/2) [(1/3) C1 + C2]
where C and C1, C2 are constants of integration.