`dy/dx = xe^(y-2x)` , i am asked to form differential equation using this equation . the ans given is
`e^-y = 0.5(e^(-2x))(x+0.5) + a`
how to get the answer?

Relevant page
<a href="/differential-equations/2-separation-variables.php">2. Separation of Variables</a>
What I've done so far
`dy/dx=xe^(y-2x)`
`dy/dx=x(e^y)/(e^(-2x))`
`e^(-y)dy = xe^(-2x)dx`
`int e^(-y)dy = int xe^(-2x)dx`
`-e^(-y) = -1/2int -2xe^(-2x)dx`
`-e^(-y) = -1/2e^(-2x)+C`

I don't believe your question says "form a differential equation" since your question is already a differential equation! Perhaps the original says "Solve the differential equation"?

Your working for the first 4 lines is correct, but your answer for the integral `int xe^(-2x) dx` is not. (A hint: It's a product, not a straightforward integral.)

This page should help: Integration by Parts. Example 5 is almost identical to the integral you need to do (actually, yours requires less steps).

X

Hello
I don't believe your question says "form a differential equation" since your question is <b>already</b> a differential equation! Perhaps the original says "Solve the differential equation"?
Your working for the first 4 lines is correct, but your answer for the integral `int xe^(-2x) dx` is not. (A hint: It's a product, not a straightforward integral.)
This page should help: <a href="http://www.intmath.com/methods-integration/7-integration-by-parts.php">Integration by Parts</a>. Example 5 is almost identical to the integral you need to do (actually, yours requires less steps).

Using Integration by Parts, put
`u = x` and `dv = e^(-2x)dx`
This gives
`du=dx` and `v=-(e^(-2x))/2`
Then
`int u dv = uv - int vdu`
means
`int xe^(-2x)dx = x(-e^(-2x)/2)-int-(e^(-2x))/2dx`
`=-xe^(-2x)/2-(e^(-2x))/4`
`=-(e^(-2x))(x/2+1/4)+C`
Putting it all together, it's
`-e^(-y) = -(e^(-2x))(x/2+1/4)+C`
`e^(-y)= e^(-2x)(2x+1)/4+C`

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