What is the differential equation for dy/dx = xe^{(y - 2x)}?

Solution:

A differential equation is an equation that relates one or more functions and their derivatives.

Given: dy/dx = xe^{(y - 2x)}

dy/dx = x(e^y / e^2x)

On rearranging it:

dy / e^y = (x / e^2x)dx

Now integrating on both the sides;

∫ (x / e^2x) dx = ∫ (dy / e^y)

⇒ ∫ (xe^-2x) dx = ∫ (dy / e^y)

Integral of RHS = -e^-y + c............. (1)

On solving LHS by uv method, we get

∫ (xe^-2x) dx = (xe^-2x / -2) - ∫ (e^-2x / -2) dx

= -1/2 (xe^-2x) - 1/4 e^-2x dx

= -1/2 e^-2x (x + 1/2) + c  ................ (2)

Equating RHS and LHS, (1) and (2)

e^-y = 1/2 e^-2x (x + 1/2) + c

Hence, solution to the differential equation, dy / dx = xe^(y - 2x) is e^-y = 1/2 e^-2x (x + 1/2) + c

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What is the differential equation for dy/dx = xe^{(y - 2x)}?

Summary:

Solution to the differential equation, dy / dx = xe^{(y - 2x)} is e^-y = 1/2 e^-2x (x + 1/2) + c