NDSolve  
NDSolve[eqns, y, {x, x_{min}, x_{max}}] finds a numerical solution to the ordinary differential equations eqns for the function y with the independent variable x in the range x_{min} to x_{max}. 
NDSolve[eqns, y, {x, x_{min}, x_{max}}, {t, t_{min}, t_{max}}] finds a numerical solution to the partial differential equations eqns. 
NDSolve[eqns, {y_{1}, y_{2}, ...}, {x, x_{min}, x_{max}}] finds numerical solutions for the functions y_{i}. 
 NDSolve[eqns, y[x], {x, x_{min}, x_{max}}] gives solutions for y[x] rather than for the function y itself.
 Differential equations must be stated in terms of derivatives such as y'[x], obtained with D, not total derivatives obtained with Dt.
 NDSolve solves a wide range of ordinary differential equations as well as many partial differential equations.
 In ordinary differential equations the functions y_{i} must depend only on the single variable x. In partial differential equations they may depend on more than one variable.
 The differential equations must contain enough initial or boundary conditions to determine the solutions for the y_{i} completely.
 Initial and boundary conditions are typically stated in the form y[x_{0}]c_{0}, y'[x_{0}]dc_{0}, etc., but may consist of more complicated equations.
 The c_{0}, dc_{0}, etc. can be lists, specifying that y[x] is a function with vector or general list values.
 Periodic boundary conditions can be specified using y[x_{0}]y[x_{1}].
 The point x_{0} that appears in the initial or boundary conditions need not lie in the range x_{min} to x_{max} over which the solution is sought.
 The differential equations in NDSolve can involve complex numbers.
 NDSolve can solve many differentialalgebraic equations, in which some of the eqns are purely algebraic, or some of the variables are implicitly algebraic.
 The y_{i} can be functions of the dependent variables, and need not include all such variables.
 The following options can be given:
 The option NormFunction>f specifies that the estimated errors for each of the y_{i} should be combined using f[{e_{1}, e_{2}, ...}].
 AccuracyGoal effectively specifies the absolute local error allowed at each step in finding a solution, while PrecisionGoal specifies the relative local error.
 If solutions must be followed accurately when their values are close to zero, AccuracyGoal should be set larger, or to Infinity.
 The setting for MaxStepFraction specifies the maximum step to be taken by NDSolve as a fraction of the range of values for each independent variable.
 Possible explicit settings for the Method option include:
 "Adams"  predictorcorrector Adams method with orders 1 through 12 
 "BDF"  implicit backward differentiation formulas with orders 1 through 5 
 "ExplicitRungeKutta"  adaptive embedded pairs of 2(1) through 9(8) RungeKutta methods 
 "ImplicitRungeKutta"  families of arbitraryorder implicit RungeKutta methods 
 "SymplecticPartitionedRungeKutta" 
  interleaved RungeKutta methods for separable Hamiltonian systems 
 With Method>{"controller", Method>"submethod"} or Method>{"controller", Method>{m_{1}, m_{2}, ...}} possible controller methods include:
 "Composition"  compose a list of submethods 
 "DoubleStep"  adapt step size by the doublestep method 
 "EventLocator"  respond to specified events 
 "Extrapolation"  adapt order and step size using polynomial extrapolation 
 "FixedStep"  use a constant step size 
 "OrthogonalProjection"  project solutions to fulfill orthogonal constraints 
 "Projection"  project solutions to fulfill general constraints 
 "Splitting"  split equations and use different submethods 
 "StiffnessSwitching"  switch from explicit to implicit methods if stiffness is detected 
 Methods used mainly as submethods include:
 "ExplicitEuler"  forward Euler method 
 "ExplicitMidpoint"  midpoint rule method 
 "ExplicitModifiedMidpoint"  midpoint rule method with Gragg smoothing 
 "LinearlyImplicitEuler"  linearly implicit Euler method 
 "LinearlyImplicitMidpoint"  linearly implicit midpoint rule method 
 "LinearlyImplicitModifiedMidpoint" 
  linearly implicit Badersmoothed midpoint rule method 
 "LocallyExact"  numerical approximation to locally exact symbolic solution 
 The setting InterpolationOrder>All specifies that NDSolve should generate solutions that use interpolation of the same order as the underlying method used. »
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