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}.
Terms {company} on non-homogeneous differential-equation right side can be similar to terms on left side.
Slight deviations from conic sections {method of perturbations} {perturbations method} can solve differential equations.
To solve differential equations with derivatives of x^n, sin(a*x), or cos(a*x), reset coefficients to one or zero, solve, and then put back coefficients {method of undetermined coefficients} {undetermined coefficients method}.
Small integral-value changes {method of variation of constants of integration} {variation of constants of integration method} can solve differential equations.
To solve differential equations with derivatives of functions that are not x^n, sin(a*x), or cos(a*x), use parameters to make ordinary differential equation and vary parameters to simplify equation {method of variation of parameters} {parameter variation method} {variation of parameters method}.
To solve differential equations with derivatives of functions that are not x^n, sin(a*x), or cos(a*x), substitute power series, such as Taylor series, for function {power series method}.
Substituting with algebraic equations can solve differential-equation systems {relaxation method, mathematics} {relaxation process}. Over an interval, select number of discrete points equal to number of variables in differential equations. At points, find approximate function values. At points, find partial derivative slope {differential coefficient} with respect to each variable: Df(x(i)) / Dx(i), where D denotes partial derivative, x is variable, and i is point/variable index. Write same number of algebraic equations as number of variables and points, each with a differential coefficient. Solve algebraic-equation system by computer.
iteration
Recognition algorithms can use iteration to move simultaneously toward optimum parameter values. Enhance some frequencies. Correlate with template. Equalize frequency histogram for more contrast. Subtract slowly varying information {background, recognition}. Find edge that has fast intensity change, using templates. Find surface orientations by neighboring reflectances. Find distances. Find velocities by comparing succeeding images. Find discontinuities and continuities.
After integration, solutions need point {boundary value, solution} {initial value, solution} to find integration constant. Problem can have no boundary or initial value {boundary value problem} {initial value problem}. Method of arithmetic means and method of sweeping out can find solutions to ordinary and partial differential-equation systems.
Potential function or harmonic function may or may not exist at boundary {Dirichlet problem} {first boundary-value problem}.
For homogeneous differential equations, equations {indicial equation} {characteristic equation, solution} can find solutions using base e raised to a power. r^n + a1 * r^(n - 1) + a2 * r^(n - 2) + ... + an = 0, where n is equation order, and r is general-solution highest power of e. Indicial equations remove highest-power term from differential equations, reducing equation degree.
Factors {integrating factor} can multiply an equation to make equation homogeneous.
To solve homogeneous differential equations, isolate variables {method of separation of variables} {separation of variables method}. Roots are e^(q*x) * (a + b*x + c * x^2 + ...), where q is coefficient, x is independent variable, and a b c are coefficients.
For first-order partial differential equations with n variables, variable separation can make ordinary differential equations with n parameters {method of characteristics} {characteristics method}. Characteristic curves and integrals are envelopes.
For first-order partial differential equations, variable separation can result in ordinary differential equations with parameters {Lagrange method}.
Power series with convergence domain can solve partial-differential-equation systems {method of majorant functions} {majorant function method}.
Green's theorem and Green's function can solve partial differential equations {method of singularities} {singularities method}.
3-Calculus-Differential Equation
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Date Modified: 2022.0225