Consider the generic third order differential equation yP2ty

Consider the generic third order differential equation y\"\'+P_2(t)y\"+P_1(t)y\'+p_0(t)y =0 with a set of solutions {y_1(t), y_2(t) y_3(t)}. Let w(t) denote the Wronskian of the solutions. According to Abel\'s Theorem, what is the first order differential equation the Wronskian solves? Rewrite the differential equation y\" +p_2(t)y\" + P_1(t)y\' + P_0(t)y = 0 as a order linear differential equations of the form y\'(t) = P(t)y(t) and identify the matrix function P(t). Let {y_1(t),y_2(t), y_3(t) a of solutions to the linear system y\'(t) = P(t)y(t) from Part (b), and W(t) be their Wronskian. Show that the differential equation satisfied by W(t) is the same as the differential equation given in part, (a).

Solution

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