Integration in differencial equations

Daniel Garla Pismel

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Can I integrate the following with respect to t:

dx/dt = (x(t) + 105)/65
x = (x(t) = 105)/65 * t

I am not sure if I can do this because x(t) is a function of time that I don't know. Can anyone help me out and explain it to me?
 

skeeter

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\(\displaystyle \frac{dx}{dt} = \frac{x+105}{65}\)

separate variables ...

\(\displaystyle \frac{dx}{x+105} = \frac{dt}{65}\)

now integrate both sides ...
 

Daniel Garla Pismel

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\(\displaystyle \frac{dx}{dt} = \frac{x+105}{65}\)

separate variables ...

\(\displaystyle \frac{dx}{x+105} = \frac{dt}{65}\)

now integrate both sides ...
Will it give the same answer as the way I did gave? If no, I didn't get it
 

Subhotosh Khan

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Will it give the same answer as the way I did gave? If no, I didn't get it
NO - it will not .

Your answer (OP) was absolutely misguided (not even wrong!)

Integrate the "expressions" derived in response #2 - tell us what you found.
 

Daniel Garla Pismel

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>Your answer (OP) was absolutely misguided (not even wrong!)

What's the point of writing like this? The only thing you achieved was making me feel bad

And the answer I've got now is x = 1,613t/(1-t/65). Is this right?
 

Jomo

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Suppose x= t^2.
Then using your method \(\displaystyle \int xdt = xt +c = x\sqrt{x} + c= x^{3/2}+ c\)

Since x=t^2, those two parallel lines between x and t^2 tells us we can replace t^2 with x and x with t^2 whenever we like.

So \(\displaystyle \int xdt = \int t^2 dt = \dfrac{t^3}{3} + c \neq x^{3/2}+ c = (t^2)^{3/2} + c = t^3\)
 

Subhotosh Khan

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>Your answer (OP) was absolutely misguided (not even wrong!)
What's the point of writing like this? The only thing you achieved was making me feel bad
And the answer I've got now is x = 1,613t/(1-t/65). Is this right?
The point was to make you review your response - and find your "mistake".

Now you found:

x = 1,613t/(1-t/65)

Differentiate "your answer" and see if it satisfies the given DE. If it does then you may be correct - if it does not you are incorrect.

At a first glance your answer is incorrect because you are missing the "constant of integration". Looks like you did not review your work again.
 

Daniel Garla Pismel

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Thank you, guys, I got it now. Thank you
 

skeeter

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Can I integrate the following with respect to t:

dx/dt = (x(t) + 105)/65
x = (x(t) = 105)/65 * t
from your original post ...

\(\displaystyle \frac{dx}{dt} = \frac{x+105}{65}\)

separate variables ...

\(\displaystyle \frac{dx}{x+105} = \frac{dt}{65}\)

integrate ...

\(\displaystyle \ln|x+105| = \frac{t}{65} + C_1\)

\(\displaystyle x+105 = e^{\frac{t}{65} + C_1}\)

\(\displaystyle x = C_2e^{\frac{t}{65}} - 105\)
 
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