# Differential equations with one variable

• Nov 2nd 2010, 08:58 AM
masterk
Differential equations with one variable
Hi!
I have some problems with solving the following differential equation:
$\displaystyle x' - x^{2} = 1$
where the unknown is $\displaystyle x(t)$
It is probably not that difficult but I am unable to find any examples involving only one unknown.
Any help would be appreciated!
• Nov 2nd 2010, 09:03 AM
Unknown008
Uh...

$\displaystyle x' = 1 + x^2$

$\displaystyle \dfrac{dx}{dt} = 1 + x^2$

$\displaystyle \displaystyle \int \dfrac{1}{1+x^2}\ dx = \int\ dt$

?
• Nov 2nd 2010, 01:45 PM
HallsofIvy
Quote:

Originally Posted by masterk
Hi!
I have some problems with solving the following differential equation:
$\displaystyle x' - x^{2} = 1$
where the unknown is $\displaystyle x(t)$
It is probably not that difficult but I am unable to find any examples involving only one unknown.
Any help would be appreciated!

I would be very surprised at that! In every differential equations text book I have ever seen, MOST of the problems in at least the first several chapters have only one unknown!
If, for example, you had an equation that said
$\displaystyle \frac{dx}{dt}- x^2= 1$
that is an equation that has "only one unknown"- x. The 'independent variable', t, is not an unknown. And, in fact, since you say that "the unknown is x(t)", the equation
$\displaystyle x'- x^2= 1$
is exactly the same as the equation
$\displaystyle \frac{dx}{dt}- x^2= 1$.

As Unknown008 said, that is a separable equation. Separate the variables and integrate.