Existence of a Unique Solution Let the coefficient functions and g(x) be continuous on an interval I and let the leading coefficient function not equal.

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Presentation transcript:

Existence of a Unique Solution Let the coefficient functions and g(x) be continuous on an interval I and let the leading coefficient function not equal 0 for every x in I. If x is any point in this interval, then a solution y(x) of the initial-value problem exists on the interval and is unique.

Homogeneous Equations Nonhomogeneous Equations

Superposition Principle— Homogeneous Equations Let be solutions of the homogeneous nth-order differential equation on an interval I. Then the linear combination where the coefficients are arbitrary constants, is also a solution on the interval.

Corollaries A constant multiple of a solution of a homogeneous linear differential equation is also a solution. A homogeneous linear differential equation always possesses the trivial solution.

Linear Dependence/Independence A set of functions is said to be linearly dependent on an interval I if there exist constants, not all zero, such that for every x in the interval. If the set of functions is not linearly dependent on the interval, it is said to be linearly independent.

Criterion for Linearly Independent Solutions Let be n solutions of the homogeneous linear nth-order differential equation on an interval I. Then the set of solutions is linearly independent on I if and only if for every x in the interval.

Fundamental Set of Solutions Let set of n linearly independent solutions of the homogeneous linear nth- order differential equation on an interval I is said to be a fundamental set of solutions on the interval. There exists a fundamental set of solutions for the homogeneous linear nth-order DE on an interval I.

General Solution-Homogeneous Equation Let be a fundamental set of solutions of the homogeneous linear nth- order differential equation on an interval I. Then the general solution of the equation on the interval is where the coefficients are arbitrary constants.