Now answered: Some Special Second Order Equations. Second order equations involve the

Chapter 2, Problem 42

(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 Independent Variable Missing. Consider second order differential equationsof the form y = f(y, y), in which the independent variable t does not appear explicitly.If we let v = y, then we obtain dv/dt = f(y, v). Since the right side of this equation depends ony and v, rather than on t and v, this equation contains too many variables. However, if we thinkof y as the independent variable, then by the chain rule, dv/dt = (dv/dy)(dy/dt) = v(dv/dy).Hence the original differential equation can be written as v(dv/dy) = f(y, v). Provided thatthis first order equation can be solved, we obtain v as a function of y. A relation between yand t results from solving dy/dt = v(y), which is a separable equation. Again, there are twoarbitrary constants in the final result. In each of 42 through 47, use this method tosolve the given differential equation.yy + (y)2 = 0