Question

Solve the initial value problem (dx/dt)-4x=cos (3t) with x(0)=0

Answer 1

There are a couple of ways to skin this cat. Remember first that for the equation without the initial condition, the general solution is
x(t) = x_h(t) + x_p(t) where x_h is the solution of the homogeneous equation and x_p is to the inhomogeneous equation above.

Hence there are three steps
1. Find the homogeneous solution
2. Find the particular solution
3. Write down the general solution and then apply the initial condition

Can you do these three steps?

Answer 2

no

Answer 3

i cant do the 1 step

Answer 4

i need help with that part

Answer 5

So the homogeneous equation is

x' - 4x = 0

Is it not clear how to solve that?

Answer 6

i.e., dx/dt = 4x

Answer 7

but is not = to 0

Answer 8

there is cos 3t

Answer 9

Like I said above, step 1 is the homogeneous equation where it IS equal to zero.

Step 2 is the inhomogeneous equation where it is equal to cos(3t)

Answer 10

ok then ?

Answer 11

So can you solve the homogeneous equation? What is that solution, x_h?

Answer 12

can u expaline more coz i miss that class so i have no idea ho to solve this kind of problem

Answer 13

here, watch this good lecture on the topic. This will get you up to speed and yes, it's worth the 50 minutes:
http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/lecture-7-first-order-linear-with-constant-coefficients/

Answer 14

actually ... only 41! A bargain.

Answer 15

ok thinks but can u solve this problem with me coz i wana make sure if i get the right unsewer

Answer 16

You should learn the theory first. You might want to go back to lecture 3 first:
http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/lecture-3-solving-first-order-linear-odes/

Answer 17

Especially if you can't solve the equation
dx/dt - 3x = 0

This is the most basic sort of ODE there is, and is dealt with in the lecture 3 I just linked to.

Answer 18

ok im kind of getting the theory and i ll try to apply on this problem but i told u i need to make sure if my unser is right or no

Answer 19

IF you can't solve dx/dt - 4x = 0, then you're not going to understand my solution to your general equation and more importantly, you will completely fail your examination.

Answer 20

i get for (dx/dt)-3x=0
e^tx(dx/dt)-3e^(-3t)y =0

Answer 21

is that right

Answer 22

?

Answer 23

r u here ?

Answer 24

So what exactly is the function x_h for dx/dt - 4x = 0? (I know I've changed the 3 to 4)

Answer 25

I'll show you

if dx/dt = 4x
then
\[\frac{dx}{x} = 4 \ dt\]

Answer 26

ok

Answer 27

Now integrate both sides to get

ln x = 4t + c

Finally, take the exp of both sides to get

x(t) = exp(4t + c)
= A.exp(4t)
where A = e^c, an arbitrary constant

I.e., the general solution of the homogeneous equation

x' - 4 x = 0

is x(t) = Ae^(4t)

Answer 28

Now for the inhomogeneous part of the solution which you need, you really, really should watch lecture 3.

Answer 29

ok

Answer 30

Let me know when you have the integrating factor for your equation.