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Real Functions in One Variable - Simple 1... - eBooks and textbooks from bookboon.com

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The equation is a first order differential equation, in which the variables can be separated, dx. Example 2.11 Find in an explicit form the complete solution (including a discussion of the domains) of the differential equation.

Figure 1: The solution for which f (0) = 1, including its asymptotes.
Figure 1: The solution for which f (0) = 1, including its asymptotes.

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Figure 13: The wanted solution through the point
Figure 13: The wanted solution through the point

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3 Linear differential equation of first order

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Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click the ad to read more Click the ad to read more Click the ad to read more Click the ad to read more Click the ad to read more Click the ad to read more. Consequently, the term tx can only be compensated by a term on the right-hand sidet2, when the polynomial x is of the first degree.

Figure 18: Graph of the solution x = − 1 2
Figure 18: Graph of the solution x = − 1 2

4 The Existence and Uniqueness Theorem and other theoret- ical considerations

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Figure 30: The graph of the derivative h  (t).
Figure 30: The graph of the derivative h (t).

5 The Bernoulli differential equation

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The equation is a nonlinear differential equation of the first order (the so-called Bernoulli differential equation), which with a rearrangement can be put into a form in which the variables can be separated. Since 2y = 0, this equation is equivalent to the first-order linear differential equation with constant coefficients. Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on ad to read more Click on ad to read more Click on ad to read more Click on ad to read more Click on ad to read more Click on ad to read more Click on ad to read more Click on ad to read more Click on ad to read more Click on ad to read more Click on ad to read more Click on ad to read more Click on ad to read more Click on the ad to read more.

Figure 33: Some solution curves.
Figure 33: Some solution curves.

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Figure 35: Sketch of some solution curves of the Bernoulli equation.
Figure 35: Sketch of some solution curves of the Bernoulli equation.

6 The setup of model equations

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Figure 36: The intersection point of the graph for p(t) = 1000 −899 exp
Figure 36: The intersection point of the graph for p(t) = 1000 −899 exp

1 C cos 2t

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Figure 38: Radius vector r(t) and the angle θ(t) for the particle P , and the orthogonal projections x(t) and y(t) onto the axes
Figure 38: Radius vector r(t) and the angle θ(t) for the particle P , and the orthogonal projections x(t) and y(t) onto the axes

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Figure 4: The solution x = t 4 for t > 0.
Figure 5: The solution x = (t 2 − 1) 2 for t > 1.
Figure 7: The curve x = exp  1
Figure 8: The solution curve x = − ln  t 2 − 1
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