Solve the differential equation. (dy)/(dx)=e^(2x-2y)

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Solve the differential equation.  (dy)/(dx)=e^(2x-2y)

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Rearrange equation for separation: We are given the differential equation (dy)/(dx) = e^(2x-2y). To solve this, we will use the method of separation of variables, which involves rearranging the equation so that all terms involving y are on one side and all terms involving x are on the other side.Integrate with respect to variables: First, we rewrite the equation to separate the variables x and y: (dy)/(dx) = e^(2x-2y) We can write this as e^(2y) dy = e^(2x) dx.Apply constant of integration: Next, we integrate both sides of the equation with respect to their respective variables. This means we will integrate e^(2y) with respect to y and e^(2x) with respect to x.Eliminate fraction: The integral of e^(2y) dy is (1/2)e^(2y), and the integral of e^(2x) dx is (1/2)e^(2x). So we have: (1/2)e^(2y) = (1/2)e^(2x) + C, where C is the constant of integration.Take natural logarithm: We multiply both sides of the equation by 2 to get rid of the fraction: e^(2y) = e^(2x) + 2C.Simplify using logarithm property: Now, we take the natural logarithm of both sides to solve for y: ln(e^(2y)) = ln(e^(2x) + 2C).Solve for y: Using the property of logarithms that ln(e^u) = u, we simplify the left side to get: 2y = ln(e^(2x) + 2C).Finally, we divide both sides by 2 to solve for y: y = (1/2)ln(e^(2x) + 2C).

Solve the differential equation. $\newline$ $\frac{dy}{dx}=e^{2x-2y}$

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Solve the differential equation. $\newline$ $\frac{dy}{dx}=e^{2x-2y}$