An ODE of the form $M(x,y)dx+N(x,y)dy=0$ is called "good" if $\frac{\partial (M(x,y))}{\partial y}=\frac{\partial (N(x,y))}{\partial x}$

We are given the differential equation $(3x^2y+2xy+y^3)dx+(x^2+y^2)dy=0$. This ODE is not "good". We are asked to find $\mu (x,y)$ such that:

$$\mu (x,y)(3x^2y+2xy+y^3)dx+\mu (x,y)(x^2+y^2)dy=0, (*)$$ is "good".

What I did:

if the equation $(*)$ is good then $\mu_y (x,y) M(x,y)+\mu (x,y)M_y (x,y)=\mu_x (x,y)N(x,y)+\mu (x,y)N_x (x,y)$

so we get

$\mu_y(x,y)(3x^2y+2xy+y^3)+\mu(x,y)(3x^2+2x+3y^2)=\mu_x(x,y) (x^2+y^2)+2x \mu(x,y)$

And now I'm stuck.

Even if we were to guess $\mu_x(x,y)=0$ or $\mu_y(x,y)=0$ we will never get something like $\frac{\mu_y}{\mu}=\phi(y)$ or $\frac{\mu_x}{\mu}=\psi(x)$. $\mu$ seems to depend on both variables and unless the above restrictions apply (which they don't here) I don't know how to find $\mu$. Please help.

  • 3
    $\begingroup$ The standard terminology for 'good' is 'exact'. $\endgroup$ – Git Gud Nov 23 '14 at 18:16
  • 2
    $\begingroup$ I'm pretty sure it is called either "exact" or "total". Never heard of "good". $\endgroup$ – UserX Nov 23 '14 at 18:17
  • $\begingroup$ Well, as long as the question was understandable... $\endgroup$ – Oria Gruber Nov 23 '14 at 18:18
  • 2
    $\begingroup$ And $\mu$ is called the 'integrating factor' $\endgroup$ – RHP Nov 23 '14 at 18:18
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    $\begingroup$ Try $\mu(x,y)=e^{3x}$. $\endgroup$ – Git Gud Nov 23 '14 at 18:21

Notice that: $$ \frac{M_y - N_x}{N} = \frac{(3x^2 + 2x + 3y^2) - (2x)}{x^2 + y^2} = 3 $$ which is independent of $y$ (and also $x$, by conincidence). This suggests that we guess that $\mu_y = 0$ so that $\mu$ is a function of $x$ only. Thus, we obtain: $$ \mu M_y = \mu_x N + \mu N_x \iff \frac{d\mu}{dx} = \frac{M_y - N_x}{N} \cdot \mu = 3\mu $$ This ODE is separable/linear and can be easily solved to obtain $\mu(x) = e^{3x}$.


Solution 1


Make the ansatz $\mu(x)$(the reason is explained in Adriano's answer)

$$(\mu (x)M(x,y))_y=(\mu (x)N(x,y))_x\iff (3x^2+2x+3y^2)\mu (x)=\mu (x)_x (x^2+y^2)+2x\mu (x)\iff \frac{\mu (x)_x}{\mu (x)}=3 \iff \mu (x)=e^{3x}$$

Solution 2


$$3x^2y+(x^2+y^2)y'+y^3+2xy=0\stackrel{y\colon =xv}{\iff} (x^2+x^2v^2)(xv'+v)+x^3v^3+3x^3v++2x^2v\;\stackrel{\text{simplify}}{\iff}\; v'=\frac{-v^3-xv^3-3v-3xv}{x(v^2+1)}\iff v'=-\frac{v(x+1)(v^2+3)}{x(v^2+1)}\iff \frac{v'(v^2+1)}{(v^2+3)v}=-\frac{x+1}{x}$$

Notation in case 1;

$X(a,b)_a\colon =\frac{\partial X(a,b)}{\partial a}$

Notation in case 2;



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