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How do you solve the separable differential equation #dy/dx = e^(x-y)#?

Answer 1

Isaiah Allen

#y = ln(e^x + C)# ; #C \equiv# constant

#dy/dx = e^(x-y)#

#dy/dx = e^xe^-y#

#(dy)/(e^-y) = e^xdx#

#e^y dy = e^x dx#

#inte^ydy = \int e^xdx#

#e^y = e^x + C# ; #C \equiv# constant

#ln(e^y) = ln(e^x + C)#

#y = ln(e^x + C)# ; #C \equiv# constant

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Answer 2

Elijah Abram

To solve the separable differential equation ( \frac{dy}{dx} = e^{x-y} ), follow these steps:

Separate the variables by moving terms involving ( y ) to one side and terms involving ( x ) to the other side.
Integrate both sides with respect to their respective variables.
Solve for ( y ) to find the general solution.

Here are the detailed steps:

[ \frac{dy}{dx} = e^{x-y} ]

Separate variables:

[ e^y dy = e^x dx ]

Integrate both sides:

[ \int e^y , dy = \int e^x , dx ]

[ e^y = e^x + C ]

Solve for ( y ) by taking the natural logarithm of both sides:

[ y = \ln(e^x + C) ]

where ( C ) is the constant of integration.

So, the general solution to the differential equation is:

[ y = \ln(e^x + C) ]

where ( C ) is an arbitrary constant.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.