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In the appendix to his book, Dennis Zill discusses a particular type of differential equation problem he provides on page 358. In this problem, the system has two variables which are both on one side of a nonzero constant. He asks what happens when one or both of the variables gets equal to zero. This topic is worth discussing because it is not always clear what will happen when a system becomes zero dimensional and how to find a solution that will satisfy a given requirement. In most cases, it will be true that a solution with a single variable will not be the same as a solution with two variables. However, this is not always the case. In most cases, a solution with one variable will be a function that approaches a horizontal asymptote as one of the variables gets closer and closer to zero. This is because only one degree of freedom is available for this system and it does not have enough information to find any sort of horizontal asymptote with no slope at all. If we use the method discussed in class to solve this problem, there are several ways to think about this. One way of thinking about it is the fact that the two equations are written in two separate ways. The first way involves dividing both sides by a nonzero constant. Another way is to use the method discussed in class to find the derivative for this differential equation system and set it equal to zero. One could also think about this situation as if it were an initial value problem with two unknowns. If one of the variables is fixed, then it becomes an equation with only one variable left to solve for. This one variable will always be on the horizontal asymptote that forms when solving differential equations like these with respect to time because there are no other variables affecting it anymore. This problem deals with non-homogeneous differential equation systems. The method of applying the two sides to the same function was developed by Gauss. It can be used to find infinite different solutions for any given set of initial conditions that will satisfy the original differential equation system. Assume that "y" is a function of x and "x". Then, this system can be written as follows: so that, for some constant "a": formula_1 formula_2 where φ("x","y") is a primitive solution of formula_3. The set of all primitive solutions of the homogeneous system can also be used to find other solutions, for any fixed value of "a". To do this, take the partial derivative with respect to "a" to get the following system: formula_4 where, again, φ("x","y") is a primitive solution of formula_5. The remaining solutions are obtained by taking derivatives with respect to each variable in turn. The procedure will terminate when formula_6 is zero or when one derivative has been taken multiple times. Differentiating the resulting equation again returns one further solution. Accordingly, there are infinitely many solutions. cfa1e77820
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