A) An equation of form y''= F(x,y') in which the dependent variable y is missing, setting w = y' (so that w' = y'') yields the pair of equations
w' = F(x,w) ,
y' = w
Once w(x) is determined, we integrate it to obtain y(x).
Using this method, solve
2xy'' - y' + 1/y' = 0
B) To solve an equation of the form y'' = F(y,y') in which the independent variable x is missing, setting w = dy/dx yields, via the chain rule,
d^2y/dx^2 = dw/dx = (dw/dy)(dy/dx) = wdw/dy
Thus, y'' = F(y,y') is equivalent to the pair of equations
1) w dw/dy = F (y, w) ,
2) dy/dx = w.
In equation (1) notice that y plays the role of the independent variable; hence, solving it yields w(y). Then substituting w(y) into (2) we obtain a separable equation that determines y(x).
Using this method, solve the following equations:
1) 2y(d^2y/dx^2) = 1 + (dy/dx)^2
2) d^2y/dx^2 + y dy/dx = 0.