Theorem of reduction of order, Mathematics

In this theorem we identify that for a specified differential equation a set of fundamental solutions will exist.

Consider the differential equation

 y′′ + p (t ) y′ + q (t ) y = 0

 Here p(t) and q(t) are continuous functions on any interval I. select t0 to be any point in the interval I. Let y1(t) be a solution to the differential equation which satisfies the initial conditions.

 y(t0) = 1

 y′ (t0) = 0

 Let y2(t) be a solution to the differential equation which satisfies the initial conditions.

 y (t0) = 0

 y′ (t0) = 1

 So y1(t) and y2(t) form a fundamental set of solutions for the differential equation.

This is easy enough to illustrate that these two solutions form a fundamental set of solutions. Just calculate the Wronskian.

976_THEOREM of Reduction of Order.png

Thus, fundamental sets of solutions will exist; we can solve the two IVP's specified in the theorem.

Posted Date: 4/10/2013 5:24:18 AM | Location : United States







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