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

Apr 14, 2018

See below.

#### Explanation:

Making the substitution

$y = \frac{u '}{u}$ and substituting we get

$u ' ' - x u = 0$

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

NOTE

A series solution can be worked-out so making

$u = {\sum}_{k = 0}^{\infty} {a}_{k} {x}^{k}$ we obtain the recurrence relationship

$\left(k + 3\right) \left(k + 2\right) {a}_{k + 3} = {a}_{k}$ with two basic sequences

$\left\{\begin{matrix}{a}_{3} = {a}_{0} / \left(2 \times 3\right) \\ {a}_{4} = {a}_{1} / \left(3 \times 4\right)\end{matrix}\right.$ etc.