See solution: Some Special Second Order Equations. Second order equations involve the

Chapter 2, Problem 41

(choose chapter or problem)

Some Special Second Order Equations. Second order equations involve the second derivativeof the unknown function and have the general form y = f(t, y, y). Usually such equationscannot be solved by methods designed for first order equations. However, there are two typesof second order equations that can be transformed into first order equations by a suitablechange of variable. The resulting equation can sometimes be solved by the methods presentedin this chapter. 36 through 51 deal with these types of equations.Equations with the Dependent Variable Missing. For a second order differential equationof the form y = f(t, y), the substitution v = y, v = y leads to a first order equation of theform v = f(t, v). If this equation can be solved for v, then y can be obtained by integratingdy/dt = v. Note that one arbitrary constant is obtained in solving the first order equation forv, and a second is introduced in the integration for y. In each of 36 through 41, usethis substitution to solve the given equationt2y = (y)2, t > 0