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Find the general solution of the given system.

dx/dt = 2x ? y

dy/dt = 4x ? 2y

Answer in this form x(t),y(t)=

User Handhand
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To find the general solution of the given system, we can use the method of elimination.

Multiplying the first equation by 2 and subtracting it from the second equation, we get:

2(dy/dt) - d/dt(2x) = 2(4x - 2y) - 2(2x - y)

Simplifying, we get:

d/dt(y - 2x) = 0

Integrating both sides with respect to t, we get:

y - 2x = C

where C is a constant of integration.

Substituting this expression for y into the first equation, we get:

dx/dt = 2x - (y - 2x) = 3x - C

This is a separable differential equation, which we can solve by separating the variables and integrating both sides:

(dx/x - (C/3)) = dt

Integrating both sides, we get:

ln|x - (C/3)| = t + K

where K is another constant of integration.

Solving for x, we get:

x(t) = (C/3) + Ae^(3t)

where A is a constant of integration.

Substituting this expression for x into the equation y - 2x = C, we get:

y(t) = 2(C/3) + Be^(3t)

where B is a constant of integration.

Therefore, the general solution of the given system is:

x(t), y(t) = (C/3) + Ae^(3t), 2(C/3) + Be^(3t)

where A and B are constants of integration, and C is a constant that determines the relationship between A and B.
User Sytech
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