# Integration in differencial equations

#### Daniel Garla Pismel

##### New member
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

##### Elite Member
$$\displaystyle \frac{dx}{dt} = \frac{x+105}{65}$$

separate variables ...

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

now integrate both sides ...

• Subhotosh Khan and HallsofIvy

#### Daniel Garla Pismel

##### New member
$$\displaystyle \frac{dx}{dt} = \frac{x+105}{65}$$

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

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

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

#### Daniel Garla Pismel

##### New member

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

##### Elite Member
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

##### Super Moderator
Staff member
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

##### New member
Thank you, guys, I got it now. Thank you

#### skeeter

##### Elite Member
Can I integrate the following with respect to t:

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

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

separate variables ...

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

integrate ...

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

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

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