Matrix Differential Equations Jacobs - Xecunet

Matrix Differential EquationsJacobsOne of the very interesting lessons in this course is how certain algebraictechniques can be used to solve differential Equations . The purpose ofthese notes is to describe how the solution(u1u2)of the Matrix equation(abcd)(u1u2)= (u1u2)will apply to the solution of the differential equa-tiona2d2ydt2+a1dydt+a0y= 0. As we will see later, the differential equationcan be rewritten in a Matrix form and then the eigenvectors and eigenvaluesof the Matrix then lead to a of Eigenvectors and EigenvaluesLetA=(abcd)and u=(u1u2). A number is said to be aneigenvalueofAif there is a nonzero vector uso thatA u= u. The vector uis saidto be aneigenvectorofA. Please note that the equation (A I) u= 0,whereI=(1 00 1)and 0=(00), is completely equivalent to the equationA u= u.

Application to Systems of Linear Differential Equations Problems involving interconnecting spring systems, tank systems and elec-tric circuits involve several unknown functions solving a system of differen-tial equations. Let’s consider, for example the problem of finding two functions x1 = x1(t)

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  System, Equations, Matrix, Jacob, Differential, Matrix differential equations jacobs

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