Question

What is the solution to the differential equation dy/dx= x-y^2 ?

Answer 1

See below.

Explanation:

Making the substitution

`y =(u')/u` and substituting we get

`u''-xu =0`

This is a linear differential equation with solution not representable with elementary functions.

NOTE

A series solution can be worked-out so making

`u = sum_(k=0)^oo a_k x^k` we obtain the recurrence relationship

`(k+3)(k+2)a_(k+3) = a_k` with two basic sequences

`{(a_3 = a_0/(2 xx 3)),(a_4 = a_1/(3 xx 4)):}` etc.