# Concept derivation guides

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Concept guide



## [Ordinary differential equation: finding the order and checking the candidates y=x² and y=x²+x](/en/derivations/derivation-mko-ordinary-differential-equation-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-ordinary-differential-equation)

On the interval I=ℝ consider the ordinary differential equation y″−2y′+y=x²−4x+2. (1) Determine its order; (2) check that y₁(x)=x² is a solution on all of ℝ (its residual is identically zero); (3) prove that y₂(x)=x²+x has residual x−2, which vanishes only at x=2, so it is not a solution on ℝ; (4) check the initial values y(0)=0, y′(0)=0 and, citing the existence and uniqueness theorem for second-order linear equations with its hypotheses, conclude that y₁ is the unique solution of this initial value problem; (5) show that without initial values the solution is not unique.

Ordinary differential equation

derivation-mko-ordinary-differential-equation-01





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