Reference no: EM13560374

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.