ODE Problem

**Bullet** · July 11th 2011, 11:21 AM

Is anyone able to do this?
Im having lots of problems

**pickslides** · July 11th 2011, 01:53 PM

For $\displaystyle \frac{d^2i}{dt^2}+400i= 0$ what is the charactersitic equation? You will need to solve this equation to contiune.

Hint: Its a quadratic

**hatsoff** · July 11th 2011, 02:17 PM

There's probably an easier way to do this, but one valid and straightforward method is to convert to the system

$\left[\begin{array}{c}x_1\\x_2\end{array}\right]'=\left[\begin{array}{cc}0&1\\-400&0\end{array}\right]\left[\begin{array}{c}x_1\\x_2\end{array}\right]$ ,

where $i(t)=x_1$ . We use this to find the fundamental matrix

$X(t)=\left[\begin{array}{cc}-\cos 20t&(\sin 20t)/20\\20i\cos 20t&i\sin 20t\end{array}\right]$

and determine $i(t)=a(-\cos 20t)+b(\sin 20t)/20$ for some constants $a,b\in\mathbb{R}$ . It's easy to verify that the solution to the IVP in (i) is then given by $i(t)=5\cos 20t$ .

**Bullet** · July 11th 2011, 02:24 PM

Hi. Thanks for the reply
I should have been more specific sorry. its part ii) that Im really stuck with. sorry
for part i) I put the values into the quadratic formula and then with the obtained values substituted them into general solution giving me the desired graph that i was looking for
If you could help me with part ii) that would be great.
thanks

**Prove It** · July 11th 2011, 09:35 PM

Originally Posted by Bullet

Hi. Thanks for the reply
I should have been more specific sorry. its part ii) that Im really stuck with. sorry
for part i) I put the values into the quadratic formula and then with the obtained values substituted them into general solution giving me the desired graph that i was looking for
If you could help me with part ii) that would be great.
thanks

For part ii) you have solved the homogeneous DE. Now solve the nonhomogeneous using variation of parameters. The general solution is the sum of the homogeneous and nonhomogeneous solutions.

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