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An example of modeling a real-world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity and air resistance. The ball's acceleration towards the ground is the acceleration due to gravity minus the deceleration due to air resistance.
Boundary value problems are similar to initial value problems.A boundary value problem has conditions specified at the extremes ("boundaries") of the independent variable in the equation whereas an initial value problem has all of the conditions specified at the same value of the independent variable (and that value is at the lower boundary of the domain, thus the term "initial" value).
Examples of differential equations; Autonomous system (mathematics) Picard–Lindelöf theorem; Peano existence theorem; Carathéodory existence theorem; Numerical ordinary differential equations; Bendixson–Dulac theorem; Gradient conjecture; Recurrence plot; Limit cycle; Initial value problem; Clairaut's equation; Singular solution ...
Differential equations play a prominent role in many scientific areas: mathematics, physics, engineering, chemistry, biology, medicine, economics, etc. This list presents differential equations that have received specific names, area by area.
Differential equations are prominent in many scientific areas. Nonlinear ones are of particular interest for their commonality in describing real-world systems and how much more difficult they are to solve compared to linear differential equations.
represent three particles with masses , distances = , and coordinates (i,j = 1,2,3) in an inertial coordinate system ... the problem is described by nine second-order differential equations. [4]: 8 The problem can also be stated equivalently in the Hamiltonian formalism, in which case it is described by a set of 18 first-order differential ...