# Differential equation $y'=\frac{1-xy}{y-x^2}$

I tried many thing but I could not find a method to solve the differential equation $$y'=\frac{1-xy}{y-x^2}$$

$$y'=-x+\frac{1-x^3}{y-x^2}$$

$Z=y-x^2$

$$Z'+2x=-x+\frac{1-x^3}{Z}$$

$$Z(Z'+3x)=1-x^3$$

Could you please give me hint which method can be used for such first order differential equations? Thanks a lot for advice and answers

• Your equation is equivalent with $$(1-xy)dx+(x^2-y)dy=0,$$ but $(1-xy)'_{y}=-x \neq 2x = (x^2-y)'_y$ Maybe we could find integration factor – Cortizol Mar 4 '13 at 21:59
• Why do you think there is an explicit solution in known functions? – GEdgar Mar 4 '13 at 22:00
• Maybe .I tried the method too but I could not find integration factor. Thanks for advice – Mathlover Mar 4 '13 at 22:04
• Check this solution. – Mhenni Benghorbal Mar 4 '13 at 22:28
• @MhenniBeghorbal I would like to learn methods to get that result – Mathlover Mar 4 '13 at 22:35

$y'=\dfrac{1-xy}{y-x^2}$

$(y-x^2)y'=1-xy$

Let $u=y-x^2$ ,

Then $y=u+x^2$

$y'=u'+2x$

$\therefore u(u'+2x)=1-x(u+x^2)$

$uu'+2xu=1-xu-x^3$

$uu'=-3xu-x^3+1$

Try to solve this ODE by Wolfram Alpha, you will discover that the general solution is implicitly expressed by $x$ and $x+\dfrac{x^3-1}{u}$ .

Since Wolfram Alpha discover that the substitution $v=x+\dfrac{x^3-1}{u}$ leads the ODE becomes a separable ODE:

Let $v=x+\dfrac{x^3-1}{u}$ ,

Then $u=\dfrac{x^3-1}{v-x}$

$u'=\dfrac{3x^2(v-x)-(x^3-1)(v'-1)}{(v-x)^2}$

$\therefore\dfrac{x^3-1}{v-x}\dfrac{3x^2(v-x)-(x^3-1)(v'-1)}{(v-x)^2}=-3x\dfrac{x^3-1}{v-x}-x^3+1$

$\dfrac{3x^2(v-x)-(x^3-1)(v'-1)}{(v-x)^2}=-3x-(v-x)$

$3x^2(v-x)-(x^3-1)(v'-1)=-3x(v-x)^2-(v-x)^3$

$(x^3-1)(v'-1)=3x^2(v-x)+3x(v-x)^2+(v-x)^3$

$(x^3-1)v'-x^3+1=3x^2(v-x)+3x(v-x)^2+(v-x)^3$

$(x^3-1)v'=x^3+3x^2(v-x)+3x(v-x)^2+(v-x)^3-1$

$(x^3-1)v'=(x+v-x)^3-1$

$(x^3-1)v'=v^3-1$

• Thanks a lot for wonderful answer. I wonder if we did not get help of wolframalpha how we could find the transform. – Mathlover Mar 5 '13 at 1:50

Equation $$zz'=-3xz-x^3+1$$ is Abel equation of the second kind. You can find here some solvable Abel equation.

Good luck with finding your equation!