# General Solution to Almost Riccati Like Equation

Consider the differential equation

$$y' = a_0(x) + a_1(x)y + a_2(x)\frac{1}{y}$$

I am attempting to find the general solution to this. One thing I can note is that the entire equation can be rewritten as

$$y' = \frac{a_2(x) + a_0(x)y + a_1(x)y^2}{y}$$

Thus allowing us to state

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

I have no idea how to progress correctly from here.

By General Solution, I mean to ask if this can be re-written as a linear ODE.

• yes! So i'm not expecting the general solution to have clean form, I'm just hoping to write it as a high order linear ODE – frogeyedpeas Dec 29 '14 at 1:25
• If $a_0 = 0$ we have $[y^2]' = 2a_2 + 2a_1 y^2$ - a linear ODE in $y^2$ – Winther Dec 29 '14 at 1:26
• there is a transformation that will riccati equation into a second order linear equation. – abel Dec 29 '14 at 1:31
• @abel I'm wondering if a modified version of that transform can tackle this one – frogeyedpeas Dec 29 '14 at 1:31

This belongs to an Abel equation of the second kind.

In fact all Abel equation of the second kind can be transformed into Abel equation of the first kind.

Let $y=\dfrac{1}{u}$ ,

Then $y'=-\dfrac{u'}{u^2}$

$\therefore-\dfrac{u'}{u^2}=a_0(x)+\dfrac{a_1(x)}{u}+a_2(x)u$

$u'=-a_2(x)u^3-a_0(x)u^2-a_1(x)u$

• I looked at the paper and states that it can eventually be converted to $$u' = u^3 + \phi(x)$$ – frogeyedpeas Jan 15 '15 at 21:01
• If we assume we have found a solution $u_1$ then using the transformation $u = u_1 + \frac{1}{e}$ the equation then becomes $$ee' = \frac{3}{2}u_1^2e^2 + \frac{3}{2}u_1e + 1$$ – frogeyedpeas Jan 15 '15 at 21:03