Question

How to simplify e^(x+(2/3)x^3)?

Answer 1

@Nurali @dan815 @ganeshie8

Answer 2

@wio @jhonyy9 @nincompoop @uri

Answer 3

it looks simplified to me.

Answer 4

\[
e^{x+\frac 23 x^3}
\]You can't simplify the exponent.

Answer 5

Okay, then. It's because I want to solve the problem about finding the general solution of xy'+(1+2x^2)y=(x^3)(e^(-x^2)) so I figured out that the integrating factor is e^(x+(2/3)x^3).

Answer 6

So do I multiply that by the whole differential equation since you said it can't be simplified more?

Answer 7

yes

Answer 8

Wait a minute. Please don't leave. Let me work it out then.

Answer 9

How do I integrate (x^3)(e^(x+(2/3)x^3-x^2))?

Answer 10

I don't think it has an elementary anti-derivative.

Answer 11

Am I doing something wrong then?

Answer 12

@Abhisar @johnweldon1993 @Preetha @paki

Answer 13

@charlotte123 @dan815

Answer 14

don't you need to divide by \(x\) first?

Answer 15

I thought that \(xy'\) is wrong since you want just \(y'\) with no coefficient.

Answer 16

How?

Answer 17

\[
y' +(x^{-1}+2x)y = x^2e^{-x^2}
\]

Answer 18

That would make your integrating factor: \[
\exp\left(\int x^{-1}+2x\;dx\right) = \exp(\ln(x)+x^2) = e^{\ln x+x^2} = xe^{x^2}
\]

Answer 19

So lets see...

\[\large xy' + (1 + 2x^2)y = (x^3)(e^{-x^2})\]
***divide everything by x to make y' by itself
\[\large \frac{dy(x)}{dx} + (\frac{1}{x} + 2x)y(x) = e^{-x^2} x^2\]

And from there it looks like the intergrating factor should be \(\large e^{\int(\frac{1}{x} + 2x)dx}\)

Answer 20

Let me try that, wio. Please don't leave.

Answer 21

yes perfect as wio has it!

Answer 22

The constant of integration would make it \(Cxe^{x^2}\).

Answer 23

Yes, I got it! The integrating factor is xe^(x^2) like wio said. I've got the right answer. Thank you so much, wio! Since you told me to divide everything by x at first, it made everything so much easier!

Answer 24

You always need to get the \(y'\) term to have a coefficient of 1.

Answer 25

It's standard form.

Answer 26

I see.