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the slope field is an array of slope marks in the phase space (in any number of dimensions depending on the number of relevant variables; for example, two in the case of a first-order linear ODE, as seen to the right). Each slope mark is centered at a point (,,, …,) and is parallel to the vector
Fig. 1: Isoclines (blue), slope field (black), and some solution curves (red) of y' = xy. The solution curves are y = C e x 2 / 2 {\displaystyle y=Ce^{x^{2}/2}} . Given a family of curves , assumed to be differentiable , an isocline for that family is formed by the set of points at which some member of the family attains a given slope .
By doing the above step, we have found the slope of the line that is tangent to the solution curve at the point (,). Recall that the slope is defined as the change in y {\displaystyle y} divided by the change in t {\displaystyle t} , or Δ y Δ t {\textstyle {\frac {\Delta y}{\Delta t}}} .
Below is a timeline of some important developments in this field. [26] [27] 1768 - Leonhard Euler publishes his method. 1824 - Augustin Louis Cauchy proves convergence of the Euler method. In this proof, Cauchy uses the implicit Euler method. 1855 - First mention of the multistep methods of John Couch Adams in a letter written by Francis Bashforth.
If we draw a graph of the logistic map (), we can observe that the slope of the tangent at the fixed point exceeds 1 at the boundary = and becomes unstable. At the same time, two new intersections appear, which are the periodic points x f 1 ( 2 ) {\displaystyle x_{f1}^{(2)}} and x f 2 ( 2 ) {\displaystyle x_{f2}^{(2)}} .
The phase portrait behavior of a system of ODEs can be determined by the eigenvalues or the trace and determinant (trace = λ 1 + λ 2, determinant = λ 1 x λ 2) of the system. [ 1 ] Phase Portrait Behavior [ 1 ]
The standard logistic function is the logistic function with parameters =, =, =, which yields = + = + = / / + /.In practice, due to the nature of the exponential function, it is often sufficient to compute the standard logistic function for over a small range of real numbers, such as a range contained in [−6, +6], as it quickly converges very close to its saturation values of 0 and 1.
In mathematics, the method of characteristics is a technique for solving partial differential equations.Typically, it applies to first-order equations, though in general characteristic curves can also be found for hyperbolic and parabolic partial differential equation.