Get answer: Equations with the Independent Variable Missing. Consider second order

Chapter 2, Problem 46

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Equations with the Independent Variable Missing. Consider second order differential equations of 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 on y and v, rather than on t and v, this equation contains too many variables. However, if we think of 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 that this first order equation can be solved, we obtain v as a function of y. A relation between y and t results from solving dy/dt = v(y), which is a separable equation. Again, there are two arbitrary constants in the final result. In each of 42 through 47, use this method to solve the given differential equation. yy (y )3 = 0