Integrating differential equation removes derivative and finds solution {general solution, differentiation}. Equations with nth derivative integrate n times. Variable power in highest-order derivative determines integration method.
integration constant
Because derivatives of constants are zero, general solutions are true to within an additive constant. General solutions are solution envelopes.
initial condition
Knowing one function value {boundary value} {initial value} or function derivative {initial condition} allows finding integration constant and so exact solution {particular solution} {singular solution}. If equation has nth derivative, n initial conditions find exact solution. Sum of general solution and particular solution is solution {complete solution}.
partial differential equations
Using variable-separation methods and/or infinite-series methods, to make ordinary differential equations, can solve partial differential equations.
conditions
To model situations that depend on conditions, use same differential equation and add integral equation to account for conditions separately.
existence proofs
To prove solution existence, demonstrate condition {Lipschitz condition}, demonstrate theorem {Cauchy-Lipschitz theorem}, or use iteration to reach solution {successive approximation method} {method of successive approximation} {existence of solutions}.
Mathematical Sciences>Calculus>Differential Equation>Methods
3-Calculus-Differential Equation-Methods
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Date Modified: 2022.0224