# Thread: acceleration, speed and distance second order differential equation

1. ## acceleration, speed and distance second order differential equation

sorry i am new to this site so havent mastered LaTex yet but question is fairly straightforward to write....

The acceleration, x'' , of an object which is moving in a straight line, is

-2x'+2+e^(-t)

where x(t) is the distance travelled (and t is time). Write down the
differential equation for x(t). Find its general solution and then that
solution which satisfies x(0) = 0, x'(0) = 0;

i really cant tell how i am meant to tackle this question

thanks

2. Originally Posted by sirellwood
sorry i am new to this site so havent mastered LaTex yet but question is fairly straightforward to write....

The acceleration, x'' , of an object which is moving in a straight line, is

-2x'+2+e^(-t)

where x(t) is the distance travelled (and t is time). Write down the
differential equation for x(t). Find its general solution and then that
solution which satisfies x(0) = 0, x'(0) = 0;
Hi sirellwood.

You have

$\displaystyle x''\ =\ -2x'+2+e^{-t}$

Integrate with respect to $\displaystyle t.$

$\displaystyle x'\ =\ -2x+2t-e^{-t}+C$

where $\displaystyle C$ is a constant. That is your differential equation for $\displaystyle x(t)$ (which you can rearrange as $\displaystyle x'+2x=2t-e^{-t}+C).$ You can also find $\displaystyle C$ immediately using the given fact that $\displaystyle x(0)=x'(0)=0.$

3. thanks. that does make sense to me. however the question is taken from a practise exam paper and in the solutions, it tells me that:
x(t) = A + Be^(-2t) + t - e^(-t)

it asks me to find the general solution but what you give in still in terms of x'? how can this be?

4. Originally Posted by sirellwood
thanks. that does make sense to me. however the question is taken from a practise exam paper and in the solutions, it tells me that:
x(t) = A + Be^(-2t) + t - e^(-t)

it asks me to find the general solution but what you give in still in terms of x'? how can this be?
The Abstractionist told you that you still had to solve the differential equation $\displaystyle x'+2x=2t-e^{-t}+C$!

5. ah thank you halls of ivy. so then do i use an integrating factor of e^(2x) on the DE that im left with?