nth order differential equations wronskian of solutions (theorem) homogenous and non homogenous forms can be applied to -- existence and uniqueness for linear solution(theorem 2) for particular solutions linear dependence of functions(the)

homogenous and non homogenous forms can be applied to --

let y₁, y₂, ..., yₙ be n solutions of the homogenous liner equation. then the linear combination y = c₁y₁ + c₂y₂ + ... + cₙyₙ is also a general solution wronskian of solutions (theorem) linear dependence of functions(the) principle of superposition for homogenous equations (theorem 1) existence and uniqueness for linear solution(theorem 2) for particular solutions

principle of superposition for homogenous equations (theorem 1)

suppose y₁, y₂, ..., yₙ are n solutions of the homogenous nth order equation. let W = W(y₁, y₂, ..., yₙ). if W = 0 then solutions are linearly dependent and of ≠ 0 solutions are independent wronskian of solutions (theorem) homogenous and non homogenous forms can be applied to -- existence and uniqueness for linear solution(theorem 2) for particular solutions principle of superposition for homogenous equations (theorem 1)

wronskian of solutions (theorem)

3.2 general solution of linear equations

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Calculus - Geometry

mmoranwkf Created by 6 mon ago

Cards in this deck(5)

nth order differential equations

let y₁, y₂, ..., yₙ be n solutions of the homogenous liner equation. then the linear combination y = c₁y₁ + c₂y₂ + ... + cₙyₙ is also a general solution

suppose that the function p₁, p₂, ..., pₙ then given n numbers b₀, b₁, ..., bₙ−₁, the nth order linear equation has a unique solution on the entire interval I that satisfies the n initial conditions y(a) = b₀, y'(a) = b₁, ..., yⁿ⁻¹(a) = bₙ−₁. says any such initial value problem has a unique solution on the whole interval I where the functions in 2 are continuous.(no other solutions with same initial values)

the n functions, f₁, f₂, ..., fₙ are said to be linearly dependent on the interval I provided there exist constants c₁, c₂, ..., cₙ not all zero such that c₁f₁ + c₂f₂ + ... + cₙfₙ on I; that is c₁f₁(x) + c₂f₂(x) + ... + cₙfₙ(x) = 0 for all x in I.(functions are ;linear;y dependent if and only if at least one of them is a combination of the others)

suppose y₁, y₂, ..., yₙ are n solutions of the homogenous nth order equation. let W = W(y₁, y₂, ..., yₙ). if W = 0 then solutions are linearly dependent and of ≠ 0 solutions are independent

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3.2 general solution of linear equations

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