Question

y’ +1/y = x
y’= dy/dx
y=?

Answer 1

Answer:

v = yx which is also y = vx

And dydx = d (vx)dx = v dxdx + x dvdx (by the Product Rule)

Which can be simplified to dydx = v + x dvdx

Using y = vx and dydx = v + x dvdx we can solve the Differential Equation.

An example will show how it is all done:

Example: Solve dydx = x2 + y2xy

Can we get it in F( yx ) style?

Start with: x2 + y2xy

Separate terms: x2xy + y2xy

Simplify: xy + yx

Reciprocal of first term: ( yx )−1 + yx

Yes, we have a function of (y/x).

So let's go:

Start with: dydx = ( yx )−1 + yx

y = vx and dydx = v + x dvdx : v + x dvdx = v−1 + v

Subtract v from both sides: x dvdx = v−1