In this problem we show how to generalize Theorem 3.2.7 (Abels theorem) to higherorder equations. We first outline the procedure for the third order equationy+ p1(t)y+ p2(t)y+ p3(t)y = 0.Let y1, y2, and y3 be solutions of this equation on an interval I.(a) If W = W(y1, y2, y3), show thatW=y1 y2 y3y1 y2 y3y1 y2 y3.Hint: The derivative of a 3-by-3 determinant is the sum of three 3-by-3 determinantsobtained by differentiating the first, second, and third rows, respectively.(b) Substitute for y1 , y2 , and y3 from the differential equation; multiply the first row byp3, multiply the second row by p2, and add these to the last row to obtainW= p1(t)W.(c) Show thatW(y1, y2, y3)(t) = c exp p1(t) dt.It follows that W is either always zero or nowhere zero on I.(d) Generalize this argument to the nth order equationy(n) + p1(t)y(n1) ++ pn(t)y = 0with solutions y1, ... , yn. That is, establish Abels formulaW(y1, ... , yn)(t) = c exp p1(t) dtfor this case.

Derivatives: Derivative is a special kindof limit. It’s a,it’s a kind of limit. The definition of the derivative can be approached in two different ways.One is geometrical (as a slopeof a curve) and the other one is physical (as a rate of change). A derivative at a point is the slope of the line which is tangent to the function at point . Function Interval [ ,] The Average rate of change in the interval , is given by: − ( ) − The rate of change is the slope of the line going through point [, ] , [, ]. Definition: The derivative of a function is denoted by: