Riccati Equation, special case I

Used to find the solution to an equation of the standard form,

`y'=F(x)+ky^2`

Using the substitution,

`y=-(u')/(ku)`

The equation becomes,

`(-(u')/(ku))'=F(x)+k(-(u')/(ku))^2`

We apply the product rule to the lhs.,

`(-(u')/(ku))'=-u'(1/(ku))'-(1/(ku))u''`

...and then apply the reciprocal rule to the first term,

`(-(u')/(ku))'=-u'(-(u')/(ku^2))'-(1/(ku))u''=(u')^2/(ku^2)-(1/(ku))u''`

Thus,

`(u')^2/(ku^2)-(1/(ku))u''=F(x)+k(-(u')/(ku))^2`

We expand the brackets,

`(u')^2/(ku^2)-(1/(ku))u''=F(x)+(u')^2/(ku^2)`

We can now subtract the common terms,

`-(1/(ku))u''=F(x)`

Lastly, on rearranging the equation, we arrive at the linear second order equation,

`u''=-F(x)ku`