hot to draw phase planes for differential equation systems

As such, it is usually acknowledged that there is no 'general theory' of partial differential equations, with specialist knowledge being somewhat divided between several essentially distinct subfields. Partly due to this variety of sources, there is a wide spectrum of different types of partial differential equations, and methods have been developed for dealing with many of the individual equations which arise. They also arise from many purely mathematical considerations, such as differential geometry and the calculus of variations among other notable applications, they are the fundamental tool in the proof of the Poincaré conjecture from geometric topology. For instance, they are foundational in the modern scientific understanding of sound, heat, diffusion, electrostatics, electrodynamics, thermodynamics, fluid dynamics, elasticity, general relativity, and quantum mechanics ( Schrödinger equation, Pauli equation, etc). Partial differential equations are ubiquitous in mathematically oriented scientific fields, such as physics and engineering.

Among the many open questions are the existence and smoothness of solutions to the Navier–Stokes equations, named as one of the Millennium Prize Problems in 2000. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity, and stability. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers.

However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. The function is often thought of as an 'unknown' to be solved for, similarly to how x is thought of as an unknown number to be solved for in an algebraic equation like x 2 − 3 x + 2 = 0. In mathematics, a partial differential equation ( PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. Again, trajectories cannot cross except at the origin.A visualisation of a solution to the two-dimensional heat equation with temperature represented by the vertical direction and color. Also, they cannot pass through the origin yet must be continuous lines- the only way that can happen is if the curve close to the origin so that a line toward the origin turns into a line away from the origin. In that case, trajectories close to the line with negative eigenvalue must be toward the origin and trajectories close to the line with positive eigenvalue must be away form the origin. The 'interesting' case is when one eigenvalue is positive and the other negative. If both eigenvalues were positive, you would have the same thing but with arrows pointing away from the origin. Essentially you just have lines through the origin with arrows pointing toward the origin. If you wanted to be very accurate, you could draw more lines close to the line with eigenvalue -2 to indicate that but probably your teacher will not require it. Flow close to the line with eigenvalue -2 must be twice as fast as that near the line with eigenvalue -1. Here, that means that trajectories must all move toward the origin. Also solutions are 'continuous' in the sense that trajectories through near by points must be similar. Except where the right side of the system is 0 (at the origin, the equilibrium point) there must be only a single trajectory through any point other than the origin. The crucial point here is the 'existance and uniqueness' theorem.

Hot to draw phase planes for differential equation systems