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Method for Solving these Systems of Differential Equations

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Write the system of differential equations that must be solved to find the desired flow line, and solve the sytem of differential equations using the methods…

  1. Let: , , and be constants, such that . The Linear Second-Order Differential Equation with constant coefficients is:

  1. Let: and compute:

  1. Substitute the functions from #2 into the differential equation from #1…

Since , If the equation above is true, then the characteristic equation must be in order for the equation to be true. The solutions to this equation are given by:

It can now be shown that the general solution to the differential equation in #1 is given by one of the following cases:

Let: and be arbitrary constants.

  1. If are real numbers, and . Then the general solution is:
  2. If , then the general solution is:
  3. If (complex non real-valued solutions), Then the general solution is:


Problem 1.

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Find the flow line for the vector field that passes through the point when . Plot the flow line for .


Begin by writing out the system of equations, based on the given information about . Let and represent and

Then, combine the equations such that is given in terms of and .

Now, find from the given , substitute the previous equation into , and re-arrange the terms algebraically until they are equal to 0.

From here, we can begin to use the method (mentioned earlier) for solving differential equations by substituting to replace all of the functions.


Referencing the Method: Since , and they are both real numbers, we can use the following equation, and begin to solve our system:

Begin with that we solved for earlier, and plug in the differential equation we just found. (Fun with Variables): To avoid repetitiveness, and add clarity in algebraic manipulation, I temporarily use and to represent certain expressions…



Now that we have an equation for and , we can find the Flow Line by using the given point when .

And now that , and are known, solve for the parametric equations of the flow line.


Problem 2

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Find the flow line  for the vector field  that passes through the point  when . Plot the flow line for .

For simplicity of notation, Let , and . Set up the system of equations:

Solve for :

To solve for with the quadratic equation, Let: , , and

Since has complex non real-valued solutions, with the form: , such that

The general solution to the differential equation is:

Find the derivative of .

Now that we have an equation for , we can find by using the set of equations from the beginning of the problem.

Next, we find and , based on the given point when , and solve the system of equations.

Now that we have found , and , we can solve for the flow line, parameterized by...