# Euler Differential Equation

• Jan 22nd 2009, 08:47 AM
LooNiE
Euler Differential Equation
Ok I have the equation:

theta^2(d2R/dtheta2)+3theta(dR/dtheta)-8R=20(theta^-3)

The condition is that R=10 when theta=1

I have worked out the equation as far as the General Solution.

The C.F i got was Ae^2t + Be^-4t

I then got a General Solution of Ae^2t + Be^-4t - 4e^-3t

How do I find the values of A and B ?
• Jan 22nd 2009, 08:53 AM
ThePerfectHacker
Quote:

Originally Posted by LooNiE
Ok I have the equation:

theta^2(d2R/dtheta2)+3theta(dR/dtheta)-8R=20(theta^-3)

The condition is that R=10 when theta=1

I have worked out the equation as far as the General Solution.

The C.F i got was Ae^2t + Be^-4t

I then got a General Solution of Ae^2t + Be^-4t - 4e^-3t

How do I find the values of A and B ?

You have the condition $\displaystyle \theta (1) = 10$.
Where is your condition $\displaystyle \theta ' (1) = ?$.
• Jan 22nd 2009, 09:01 AM
LooNiE
Quote:

Originally Posted by ThePerfectHacker
You have the condition $\displaystyle \theta (1) = 10$.
Where is your condition $\displaystyle \theta ' (1) = ?$.

R is finite as theta tends to infinity.
• Jan 22nd 2009, 09:06 AM
ThePerfectHacker
Quote:

Originally Posted by LooNiE
R is finite as theta tends to infinity.

Okay. You said you got the general solution of $\displaystyle Ae^{2\theta} + Be^{-4\theta} - 4e^{-3\theta}$.
For this to stay bounded as $\displaystyle \theta \to \infty$ you need $\displaystyle A=0$.
Do you see why?
• Jan 22nd 2009, 09:08 AM
LooNiE
Quote:

Originally Posted by ThePerfectHacker
Okay. You said you got the general solution of $\displaystyle Ae^{2\theta} + Be^{-4\theta} - 4e^{-3\theta}$.
For this to stay bounded as $\displaystyle \theta \to \infty$ you need $\displaystyle A=0$.
Do you see why?

yes i see. but how do i find B?
• Jan 22nd 2009, 09:16 AM
ThePerfectHacker
Quote:

Originally Posted by LooNiE
yes i see. but how do i find B?

Once you let $\displaystyle A=0$ you are let with $\displaystyle Be^{-4\theta} - 4e^{-3\theta}$.
You are told that this expression is equal to $\displaystyle 10$ with $\displaystyle \theta = 1$.
Thus, $\displaystyle Be^{-4} - 4e^{-3} = 10$.

Now solve for $\displaystyle B$.