Solving 2nd Order Diff EQ: G(x), Joe Needs Help

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In summary, the conversation is about obtaining a general solution for a second order differential equation and then finding a specific solution that remains finite as t tends to infinity. The general solution is G(x)= Ae^2x + Be^-x + cost - 3sint, and the specific solution is x=(3-B)e^2t +Be^-t +cost -3sint. The problem is that as t tends to infinity, e^2t will tend to infinity, making the solution non-finite. The solution is to find a value for B that will eliminate the e^2t term so that the solution remains finite.
  • #1
josephcollins
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Can anyone offer some advice on this problem:

Obtain a general solution for the second order differential equation:

(d^2x/dx^2) - (dx/dt) - 2x = 10sint

I obtained the general solution and now need to determine the "solution which remains finite as t tends to infinity for which x=4 at t=0. Could anyone suggest how I may approach this


My general solution was:

G(x)= Ae^2x + Be^-x + cost - 3sint

Thanks for any help,
Joe
 
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  • #2
Sorry,they have to have th same variable.It's either "t" or "x",make up your mind.Else it would have to be a PDE.

I think u can reject the positive exponential for obvious reasons...


Daniel...
 
  • #3
Well, one problem is you are confusing your dependent and independent variables!

x(t)= Ae2t+ Be-t+ cos(t)- 3 sin(t).

Now just substitute t= 0, x= 4 to get one equation in the two unknowns A and B.

Now what happens to e2t and e-t as t goes to infinity?
(sin(t) and cos(t) remain finite, of course). What do you need to do to make sure your solution doesn't go to infinity?
 
  • #4
ok, I have my general solution:

x: Ae^2t + Be^-t +cost -3sint

putting in x=4 and t=0 I obtain

4=A+B+1

so 3=A+B and A=3-B

So My final solution is:

x=(3-B)e^2t +Be^-t +cost -3sint

Is this correct, could someone verify? How about the fact that t tends to infinity, does this alter my answer at all?
 
  • #5
The problem specifically asks for the solution bounded at infinity.So that should give an idea about the value of B.

Daniel.
 
  • #6
The problem I have now is seeing what the equation does as t tends to infinity, e^-t will tend to zero, but e^2t will just tend to infinity while cost and sint are periodic, could you help me with this please?

Joe
 
  • #7
josephcollins said:
The problem I have now is seeing what the equation does as t tends to infinity, e^-t will tend to zero, but e^2t will just tend to infinity while cost and sint are periodic, could you help me with this please?

Joe


You've already found the problem and you already know what's going to happen as t goes to infinity. . e^(2t) is tending toward infinity as t goes to infinity. The problem specifies that the solution must remain finite as t goes to infinity, so you can't leave the e^(2t) in there, now can you? If you do, you're going to end up with a non-finite solution as t goes to infinity. What can you do with the coefficient B so that the problem term is no longer a factor (i.e. disappears)?

--J
 

1. What is a 2nd order differential equation?

A 2nd order differential equation is a mathematical equation that involves a function, its first derivative, and its second derivative. It is used to describe the relationship between a function and its rate of change.

2. How do I solve a 2nd order differential equation?

To solve a 2nd order differential equation, you can use various methods such as separation of variables, substitution, and the method of undetermined coefficients. It is important to identify the type of equation and use the appropriate method.

3. What is G(x) in the context of solving a 2nd order differential equation?

G(x) represents the function in the differential equation. It is the dependent variable and its value is determined by the independent variable x and its derivatives.

4. Why does Joe need help with solving the 2nd order differential equation?

Joe may need help because solving 2nd order differential equations can be a complex process and requires a strong understanding of calculus and algebraic concepts. It is always a good idea to seek assistance from a tutor or teacher if you are struggling with solving these types of equations.

5. What are some real-life applications of 2nd order differential equations?

2nd order differential equations have many applications in science and engineering, such as modeling the motion of a mass on a spring, predicting the growth of a population, and describing the behavior of electrical circuits. These equations are also used in physics, chemistry, economics, and other fields to study various phenomena.

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