MA 36600 LECTURE NOTES: MONDAY, FEBRUARY 16Second Order Differential EquationsLinear Equations.We briefly recall some facts about differential equations which we have seen over thepast few weeks. Recall that we use the notationy(n)=dnydtnto denote thenth derivative. We say that an equation of the formFt, y, y(1), . . . , y(n)= 0is annth order differential equation. In particular, we can express the highest order derivative in terms ofthe lower order derivatives:dnydtn=Gt, y, y(1), . . . , y(n−1)for some functionG(t, y1, y2, . . . , yn). Whenn= 2, we call an equation of the formd2ydt2=Gt, y,dydtasecond order differential equation.Recall that a first order differential equation is said to be a linear equation if it is in the formdydt=G(t, y)whereG(t, y) =g(t)−p(t)yfor some functionsp(t) andg(t).Similarly, we say that a second order differential equation is alinearequationif it is in the formd2ydt2=Gt, y,dydtwhereG(t, y1, y2) =g(t)−q(t)y−p(t)dydtfor some functionsp(t),q(t
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