Using D'Alembert's Principle to Find the Fundamental System of an ODE System

[mathrocks]

Hey guys, I'm stuck on this problem! I have no idea how to even go about solving it. I tried searching the net for D'Alembert's principle but nothing was helpful. Any suggestions on how to go about solving it will be much appreciated!

Homework Statement

Find with the aid of D’Alembert’s reduction principle the Fundamental system of the ODE system.

x₁(t)'= x₁(t)/t - x₂(t)

x₂(t)'= x₁(t)/t²+ 2x₂(t)/2

for which a particular solution x(t)=(t², -t) is known.

 

[lippyka]

Homework EquationsD'Alembert's reduction principle states that if a particular solution of an ODE is known, then the general solution can be found by subtracting the particular solution from the right-hand side of the equation. The Attempt at a SolutionFirst, we need to rewrite the equation in standard form:x1(t)' - x1(t)/t + x2(t) = 0 x2(t)' - x1(t)/t2 - 2x2(t)/2 = 0Then, we can use D'Alembert's reduction principle and subtract the particular solution from the right-hand side of the equation:x1(t)' - x1(t)/t + x2(t) - t2 = 0 x2(t)' - x1(t)/t2 - 2x2(t)/2 + t = 0Now, the Fundamental system of the ODE system is given by:x1(t)= c1e^(t^2/2) x2(t)= c2te^(t^2/2)