## differential equation and initial value

1. The problem statement, all variables and given/known data

[img=http://img3.imageshack.us/img3/3417/questionec.th.jpg]

2. Relevant equations

3. The attempt at a solution

I tried dividing both sides by (x^5+1), however the integration becomes really complex... can someone give me suggestions on how to do this?
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 Quote by -EquinoX- I tried dividing both sides by (x^5+1), however the integration becomes really complex... can someone give me suggestions on how to do this?
put the -10x4y on the left side then divide by tr x5+1. Integrating factor it.
 that's what I did and then I need to integrate -10x^4/(x^5+1) dx right and then then to get the integrating factor is just e to the power of whatever the result of the integration is... however the integration is quite hard...

Recognitions:
Homework Help

## differential equation and initial value

 Quote by -EquinoX- that's what I did and then I need to integrate -10x^4/(x^5+1) dx right and then then to get the integrating factor is just e to the power of whatever the result of the integration is... however the integration is quite hard...
alright now, so

$$\int \frac{-10x^4}{x^5+1} dx$$

see how d/dx(x5+1)=5x4 ?

Can you use a substitution to make this integral easier?

 Quote by rock.freak667 alright now, so $$\int \frac{-10x^4}{x^5+1} dx$$ see how d/dx(x5+1)=5x4 ? Can you use a substitution to make this integral easier?
Okay say I solve the integral and then the integrating factor would be e to the power of this resulting integral right? so then what do I need to do next in order to solve for this problem?

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 Quote by -EquinoX- Okay say I solve the integral and then the integrating factor would be e to the power of this resulting integral right? so then what do I need to do next in order to solve for this problem?

right so for y'+P(x)y=Q(x), when you multiply by an integrating factor 'u', the left side becomes d/dx(uy) . That's why we multiply by u in the first place.

So you'll need to basically integrate uQ(x) w.r.t. x
 and then divide that by the integrating factor right?

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 Quote by -EquinoX- and then divide that by the integrating factor right?
yes you can if you feel the need to.
 what do you mean by if I need to? doesn't it always works like that?
 this now comes to: $$\int \frac{x^2+5x-4}{x^5+1} e^{-2ln(x^5+1)} dx$$ I guess this can be simplified to: $$\int \frac{x^2+5x-4}{(x^5+1)^3} dx$$ is this true? how can I solve this such complex integration?

Recognitions:
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 Quote by -EquinoX- anyone please?
Re do your integrating factor as it was to integrate 10x4/(x5+1) not with the -ve sign.
 Integrating $$\frac{10x^4}{(x^5+1)}$$ the result I got is $$2ln(x^5+1)$$ and so the integrating factor is $$e^{2ln(x^5+1)}$$ which simplifies to $$(x^5+1)^2$$. Then I do $$\int (x^2+5x-4)(x^5+1)$$ and the result of this integration I divide by $$(x^5+1)^2$$ which is the integrating factor. Is this the correct step to find the solution? Please correct me if I am wrong.

Recognitions:
Homework Help
 Quote by -EquinoX- Integrating $$\frac{10x^4}{(x^5+1)}$$ the result I got is $$2ln(x^5+1)$$ and so the integrating factor is $$e^{2ln(x^5+1)}$$ which simplifies to $$(x^5+1)^2$$. Then I do $$\int (x^2+5x-4)(x^5+1)$$ and the result of this integration I divide by $$(x^5+5)^2$$ which is the integrating factor. Is this the correct step to find the solution? Please correct me if I am wrong.

yes just integrate and divide, also don't forget the constant of integration

 Quote by rock.freak667 yes just integrate and divide, also don't forget the constant of integration
by the constant of integration you mean the C right? The terms after integration that I found is very long.....