differentials

**gracey** · March 18th 2009, 07:54 AM

A point moves on the x -axis so that its coordinate at time t satisfies the differential equation dx/dt = 5 + acos2t for some value of a. it is observed that x=3 when t=0 and x=0 when t=1/4π. Find the value of a and the value of x when t=1/3π.

i am so stuck any help would be much appreciated thanks

**Grandad** · March 18th 2009, 08:20 AM

Hello gracey

Originally Posted by gracey

A point moves on the x -axis so that its coordinate at time t satisfies the differential equation dx/dt = 5 + acos2t for some value of a. it is observed that x=3 when t=0 and x=0 when t=1/4

Originally Posted by gracey

π. Find the value of a and the value of x when t=1/3π.

i am so stuck any help would be much appreciated thanks

Do you know how to work out $\int(5+a\cos2t)dt$ ? This will give you a formula for $x$ in terms of $t$ . (Don't forget $c$ , the constant of integration.)

Then plug in your two pairs of values of $x$ and $t$ , to work out the values of $a$ and $c$ . Then you can find $x$ when $t = \tfrac{1}{3}\pi$ .

Can you do it now?

Grandad

**gracey** · March 18th 2009, 08:22 AM

thanks but i'm not sure how to integrate the equation

**Grandad** · March 18th 2009, 08:38 AM

Hello gracey

$\int(5+a\cos 2t)dt = 5t -\tfrac{1}{2}a\sin 2t + c$

Grandad

**gracey** · March 18th 2009, 10:04 AM

thankyou so much, you are amazing

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