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# dy/dx = xe^(y-2x), form differntial eqaution [Solved!]

### My question

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

### Relevant page

2. Separation of Variables

### 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

X

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

## Re: dy/dx = xe^(y-2x), form differntial eqaution

Hello

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).

## Re: dy/dx = xe^(y-2x), form differntial eqaution

It seems grabbitmedia has disappeared. Anyone else like to have a go at finishing this?

X

It seems grabbitmedia has disappeared. Anyone else like to have a go at finishing this?

## Re: dy/dx = xe^(y-2x), form differntial eqaution

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

X

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

## Re: dy/dx = xe^(y-2x), form differntial eqaution

Looks good to me, stephenB!

X

Looks good to me, stephenB!