Question

Use the elimination method to find a general solution for the given linear system, where differentiation is with respect to t.

x' = 2x-y
y' = 2y-16x
Eliminate x and solve the remaining differential equation for y. Choose the correct answer below.
A. y(t) = C₁e 2t+C₂ e 6t -
OB. y(t)=C₁e-2t+C₂ e - 6t
OC. y(t)=C₁et+C₂te 6t
OD. y(t)=C₁e 2+ C₂te -21
OE. The system is degenerate.
Now find x(t) so that x(t) and the solution for y(t) found in the previous step are a general solution to the system of differential equations. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. x(t)=C₁e - 21 61

Answer 1

To solve the given system of differential equations using the elimination method, we need to eliminate one of the variables in order to solve for the remaining variable and obtain a general solution. After eliminating x, we obtain a general solution for y as y(t) = C1e^8t + C2.

Step-by-step explanation:

To solve the given system of differential equations using the elimination method, we need to eliminate one of the variables (x or y) in order to solve for the remaining variable and obtain a general solution.

We can start by multiplying the first equation by 16 and the second equation by 2 in order to make the coefficients of y in both equations the same. This gives us:

16x' = 32x - 16y
2y' = 4y - 32x

Next, we can subtract the second equation from the first equation to eliminate x:

16x' - 2y' = 32x - 4y - (4y - 32x)

Simplifying the equation gives us:

16x' - 2y' = 0

Now we can solve this differential equation for y:

2y' = 16x' => y' = 8x'

Taking the integral of both sides gives us:

y = 8x + C

So the general solution for y is y(t) = C1e^8t + C2, where C1 and C2 are arbitrary constants.