# Thread: differential equations problem

1. ## differential equations problem

I haven't dealt with differential equations in a while so I completely forget how to do this problem.

Given dx/dt=y and dy/dt=x

Verify that f(t)=Ce^t+De^-t and g(t)=Ce^t - De^-t give a solution to the system of differential equations, where C and D are arbitrary constants.

Thanks!

2. Originally Posted by morganfor
I haven't dealt with differential equations in a while so I completely forget how to do this problem.

Given dx/dt=y and dy/dt=x

Verify that f(t)=Ce^t+De^-t and g(t)=Ce^t - De^-t give a solution to the system of differential equations, where C and D are arbitrary constants.

Thanks!
Have уou to solve a system of diff. equations $\displaystyle \left\{ \begin{gathered}\frac{{dx}}{{dt}} = y, \hfill \\\frac{{dy}}{{dt}} = x. \hfill \\ \end{gathered} \right.$ ?

Differentiate, for example, the first equation $\displaystyle \frac{{{d^2}x}} {{d{t^2}}} - \frac{{dy}}{{dt}} = 0$ and because $\displaystyle \frac{dy}{dt} = x$
you get a simple diff. equation $\displaystyle x''\left( t \right) - x\left( t \right) = 0$

I think you know what you need to do next.

3. Actually, I'm having some trouble differentiating x''-x=0. I can't quite seem to get the e's in my equation.

4. Originally Posted by morganfor
Actually, I'm having some trouble differentiating x''-x=0. I can't quite seem to get the e's in my equation.
$\displaystyle \frac{d^2 x}{dt^2} - x = 0$ is a linear second order differential equation with constant coefficients. Have you studied that type of DE before?

5. I'm sure I have but it's been atleast 3 years I just need a little refresher

6. Originally Posted by morganfor
I'm sure I have but it's been atleast 3 years I just need a little refresher
Have you tried using Google to get that refresher? Keywords:

linear second order differential equation constant coefficients

7. yes i did actually - i forgot to add that to the thread earlier.

thanks everyone!

8. Originally Posted by morganfor
yes i did actually - i forgot to add that to the thread earlier.

thanks everyone!
Morganfor, the differential equations videos at Khan Academy are excellent if you're looking for a refresher. Good luck

9. This problem does not ask you to solve the equations, just show that the given functions satisfy the equations. That's much simpler.
If $\displaystyle x(t)=Ce^t+De^{-t}$ and $\displaystyle y(t)=Ce^t - De^{-t}$, what are dx/dt and dy/dt?

10. i apologize that i'm still stuck on this problem!

as far as verifying that the given solutions are correct - I'm sure I'm missing a simple step but I can't seem to differentiate f(t)=Ce^t+De^-t and g(t)=Ce^t-De^t to give me y and x. I'm confused as to which substitutions I need to make...