Solve Riccati equation dy/dx +p(x)y=g(x)y^2 +h(x) and the transformation is y=Y+1/u?

Question

and obtain for u the linear ODE u' + (2Yg - p )u= -g

Answer 1

it equals 2.

2 points

for me!

Answer 2

Substituting with u is just arithmetic, but once you get u'+(2Yg-p)u = -g is when you actually solve it.

A first order differential equation in that form can be solved by integrating factor. If the equation is in the form u' + pu = g, then the integrating factor can be shown to be given by e^(-int(p)) with respect to the independent variable.

Then you have that:

derivative(integrating factor * u) = integrating factor * g

Integrate both sides

integrating factor * u = int(integrating factor * g) + C

Just plug in with the functions you have to solve for it.