How Do You Calculate Specific Values for Constants in Differential Equations?

  • Thread starter hbomb
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In summary: You solve for A and C.In summary, the general solution of the set of equations is x=3C1e^7t-3C2e^t-C3e^-4t, y=6C2e^t, and z=2C1e^7+8C2e^t+3C3e^-4t. To find the solution at t=0, you plug in t=0 to all the t's in the general solutions and set them equal to the given point (x,y,z)=(2,3,-1). Then, solving for the constants A, B, and C will give you the specific solution.
  • #1
hbomb
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Give the general solution of the set of equations below:

x'=5x-2y+3z
y'=y
z'=6x+7y-2z

Which I found to be:
x=3C1e^7t-3C2e^t-C3e^-4t
y=6C2e^t
z=2C1e^7+8C2e^t+3C3e^-4t

Here's where I'm stuck. They want me to find the solution at t=0, (x,y,z)=(2,3,-1)

Which the professor hasn't told us how to do this.
 
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  • #2
I'm sure you know how to do this. If t=0, what is x(0)? There are two ways to express this... one is from the equation for x you solved for, the other is from the point you were given. Since t=0, all those nasty exponentials go away, and you have a system of three variables with constant coefficients (which is remarkably simpler than the initial set of differential equations you solved)
 
  • #3
ok, so what I understand from this is that I plug t=0 into all the t's of the general solutions.

x=3C1-32-C3
y=6C2
z=2C1+8C2+3C3

And then do I set these equal to (x,y,z)=(2,3,-1). What is the solutions form suppose to look like?
 
  • #4
Let's call [tex]C_1 C_2[/tex] and [tex]C_3[/tex] A, B, and C to make it easier.

You know
x(0)=3A - 3B - C=2
y(0)=6B = 3
z(0)=2A + 8B + 3C=-1

So you should be able to solve for A, B and C as numbers. For example, if 6B=3, B=2. Now you know 3A-6-C = 2, and 2A + 16 + 3C = -1
 

Related to How Do You Calculate Specific Values for Constants in Differential Equations?

What does it mean to find a solution at a point?

Finding a solution at a point means determining the value of a variable or set of variables that satisfies a given equation or system of equations at a specific point or set of points.

Why is finding a solution at a point important in science?

Finding a solution at a point is important in science because it allows us to make accurate predictions and model complex systems. It also helps us understand the relationships and interactions between different variables in a system.

What methods can be used to find a solution at a point?

There are several methods that can be used to find a solution at a point, including substitution, elimination, and graphing. These methods involve manipulating equations and variables to isolate the unknown value.

What are the potential challenges in finding a solution at a point?

Some potential challenges in finding a solution at a point include complex equations or systems, multiple solutions, and the need for precise and accurate data. In some cases, it may also be necessary to use advanced mathematical techniques to find a solution.

How can finding a solution at a point be applied in real-world situations?

Finding a solution at a point has many real-world applications, such as in physics, engineering, economics, and computer science. It can be used to optimize processes, make predictions, and solve problems in a variety of fields.

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