How Do You Calculate This Indefinite Integral in a First Order Linear ODE?

[Naincy]

Hi

How do you calculate the following indefinite integral:

$$\int \frac{2x}{125+3t} dt$$

a step by step solution would be appreciated

 

[MarkFL]

Hello and welcome to MHB! :D

I have moved your post to its own thread in our Calculus forum so it will get more prompt attention. New questions should be posted in a new thread in the most appropriate sub-forum.

Now, is it possible you mean:

$$\int\frac{2t}{125+3t}\,dt$$

If so, can you tell us what you have tried so we know where you are stuck?

 

[Naincy]

Hi. Thanx

No, it is actually 2x and not 2t in the numerator.
It is a function in two variables x and t. I need to integrate it with respect to t. Note that x cannot be considered a constant here.
so , it becomes integration of a function in two variables. i don't know how to start with it.
any help would be appreciated.

- - - Updated - - -

Hi. Thanx

No, it is actually 2x and not 2t in the numerator.
It is a function in two variables x and t. I need to integrate it with respect to t. Note that x cannot be considered a constant here.
so , it becomes integration of a function in two variables. i don't know how to start with it.
any help would be appreciated.

Basically, I need to find the solution in x for the following differential equation:
dx/dt = 1 - { 2x/(125 + 3t) }

 

[MarkFL]

Basically, I need to find the solution in x for the following differential equation:
dx/dt = 1 - { 2x/(125 + 3t) }

Oh, okay, that's a different problem entirely, and I can actually help you here. Let's first write the ODE in standard linear form:

$$\frac{dx}{dt}+\frac{2}{3t+125}x=1$$

Now, we need to compute the integrating factor $$\mu(t)=\exp\left(\int\frac{2}{3t+125}\,dt\right)$$

What do you find?

I am going to move and re-title this thread. :D

 

[Naincy]

Now, we need to compute the integrating factor $$\mu(t)=\exp\left(\int\frac{2}{3t+125}\,dt\right)$$

What do you find?

I am going to move and re-title this thread. :D

okay! right. So , i solved this linear first order differential equation and i got

x(t) =

$$\frac{125+3t}{2}\left[1+c({125+3t})^{-5/3}\right]$$

Can you just check if it's correct? Thanx. :)

 

[Naincy]

Integrating factor,here, is: (125+3t)^(2/3)

 

[MarkFL]

It's close, but not quite correct. If you show your work, we can figure out where the small error is. :D

 

[Naincy]

Did I get the integrating factor right ?

 

[I like Serena]

Did I get the integrating factor right ?

Yes.

 

[Naincy]

ok. figured out the mistake. :)

The final answer is:

x(t) = $$\frac{125+3t}{5} + \frac{c}{({125+3t})^{2/3}}$$

I think this is correct. yeah ?

 

[I like Serena]

ok. figured out the mistake. :)

The final answer is:

x(t) = $$\frac{125+3t}{5} + \frac{c}{({125+3t})^{2/3}}$$

I think this is correct. yeah ?

Yeah. ;)

 

[MarkFL]

Yes, that's correct. (Yes)