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Integrate e^(x*y) dx using a substitution

Full differential equation:

x*(dy/dx) + y = e^(x*y)

=> d(xy)/dx = e^(x*y)

Then integrate, using a substitution.

:confused:

Reply 1

Ewan

Integrate both sides wrt x & you get...

xy = integral e^(x*y) dx

Sub u=x*y & work out du/dx etc, see what you get. You'll end up with some cancellations.

(edited 12 years ago)

Reply 2

gff

I would suggest substitute

u = xy

into the differential equation.

\displaystyle \Rightarrow \frac{du}{dx} = y + x\frac{dy}{dx}

\displaystyle \therefore \ \frac{du}{dx} = e^u

You clearly cannot integrate the way you suggested! This makes no sense. :tongue:

Reply 3

Dragonfly9093

Original post by gff

I would suggest substitute

u = xy

into the differential equation.

\displaystyle \Rightarrow \frac{du}{dx} = y + x\frac{dy}{dx}

\displaystyle \therefore \ \frac{du}{dx} = e^u

You clearly cannot integrate the way you suggested! This makes no sense. :tongue:

I'm sorry, but I'm quite confused by this. :confused:

In this sense that I already found this and have tried to proceed with this substitution...

Reply 4

gff

Original post by Dragonfly9093

I'm sorry, but I'm quite confused by this. :confused:

In this sense that I already found this and have tried to proceed with this substitution...

Which part would you like me to clarify? :tongue:

Reply 5

Dragonfly9093

Original post by gff

Which part would you like me to clarify? :tongue:

Do you mean the question makes no sense or the substitution makes no sense?

Reply 6

gff

Original post by Dragonfly9093

Do you mean the question makes no sense or the substitution makes no sense?

I don't see how you can compute

\displaystyle \int e^{x \cdot y(x)}\ dx

when you are trying to find

y(x)

. The approach does not seem appropriate to me.

They way I suggested will lead you to the right answer.

Reply 7

Dragonfly9093

Original post by gff

I don't see how you can compute

\displaystyle \int e^{x \cdot y(x)}\ dx

when you are trying to find

y(x)

. The approach does not seem appropriate to me.

They way I suggested will lead you to the right answer.

d(xy)/dx = e^(x*y)
xy = integral e^(x*y) dx
xy = u
du/dx = xdy/dx + y = e^(x*y)=e^u
dx= du/e^u
u = integral e^u /e^u du
u = integral 1 du
u = u + c

Boundary conditions y=1, x=1
xy = xy + c
1 = 1 + c
xy = xy
So that just seems pointless.
Wasn't that the way you were suggesting? I substituted u = xy into the differential equation.