differential Eq, how do I make it Exact? (using an integrating factor?)

Question

$$ye^xdx-(4y+3e^x)dy=0$$

$$(\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}) * \frac{1}{M}$$ $$e^{\int(4/y)}=y^4 $$

• when multiplying $$y^4$$ though the problem still does not become exact?

attempt (cant find my error) , this will not make it exact...? what am I doing wrong?

Answer 1

Hint: Denominator should be $−M$, then the integrating factor is $$\dfrac{1}{y^4}$$

Answer 2

$$ye^xdx-(4y+3e^x)dy=0$$ $$Pdx+Qdy=0$$

Facteur integrating is $$\ln ( \mu)=-\int \frac 1 P (\partial_y P-\partial_x Q)dy=-\int \frac {4}y=-4\ln y$$ $$\implies \mu =y^{-4}$$ multiply by $1/y^4$ $$\implies \frac {e^x}{y^3}dx-(\frac 4{y^3}+3\frac {e^x}{y^4})dy=0$$ Now it's exact $$\partial_y(\frac {e^x}{y^3})=\partial_x(-\frac 4{y^3}-3\frac {e^x}{y^4})$$