# Can a third degree'Riccati' like differential equation be re-written as a third order linear ODE

Consider the equation

$$y' = a_0(x) + a_1(x)y + a_2(x)y^2 + a_3(x)y^3$$

Does there exist a substition $y = f(a_0, a_1, a_2, a_3, u, u', u'')$ such that after simplifying from this subsitution we get an equation of the form

$$u''' = b_0(x) + b_1(x)u + b_2(x)u' + b_3(x)u''$$

My Motivation for why such an substitution should exist even if non-elementary

The solution to a first order differential equation takes on the form $y = F(x,C)$ for some function F and a constant factor. This particular differential also happens to be a third degree polynomial meaning that there exist 3 unique first order equations that can be generated from it if we solve y as a function of y' and the coefficients.

nth order linear differential equations can vary by up to n constants in their solution and therefore a 3rd order linear differential equation should have enough degrees of freedom to model any third-degree polynomial differential equation such as the one I have given.

This is evidenced by the fact that second degree polynomial differential equations (THAT IS RICCATI EQUATIONS) are known to be convertible to second degree linear differential equations.

• I don't accept your logic here. Yes nth order linear differential equation is completely determined by it's nth order of characteristic polynom. It does not mean anything for nonlinear differential equation. Yes in some quite limited cases you can find a transformation of nonlinear to linear, but this is not the case IMHO.Nevertheless here a paper "irena-nesterova.livejournal.com/9536.html" – Alexander Vigodner Jun 9 '14 at 22:04
• @Alexander Vigodner, the link doesn't seem to be working – frogeyedpeas Jun 10 '14 at 1:33
• For the record, this equation is known as Abel's differential equation. – Ben Derrett Feb 27 '15 at 15:30