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ivanzaharov [21]
3 years ago
12

Solve the equation

7D%29%20dx%20%2B%20%282%20x%5E%7B6%7D-15x%20y%5E%7B6%7D%20%29dy%3D0%20%20" id="TexFormula1" title="(12 x^{5}y+12 x^{6}y-3 y^{5} -18x y^{5}) dx + (2 x^{6}-15x y^{6} )dy=0 " alt="(12 x^{5}y+12 x^{6}y-3 y^{5} -18x y^{5}) dx + (2 x^{6}-15x y^{6} )dy=0 " align="absmiddle" class="latex-formula"> leaving your answer in implicit form.
Mathematics
1 answer:
sweet-ann [11.9K]3 years ago
3 0
\underbrace{(12x^5y+12x^6y-3y^5-18xy^5)}_M\,\mathrm dx+\underbrace{(2x^6-15xy^4)}_N\,\mathrm dy=0

M_y=12x^5+12x^6-15y^4-90xy^4
N_x=12x^5-15y^4

-\dfrac{N_x-M_y}N=6\implies\mu(x)=\exp\left(\displaystyle\int6\,\mathrm dx\right)=e^{6x}

\underbrace{(12x^5y+12x^6y-3y^5-18xy^5)e^{6x}}_{\mu M}\,\mathrm dx+\underbrace{(2x^6-15xy^4)e^{6x}}_{\mu N}\,\mathrm dy=0

You can verify that the partial derivatives are equal.

F_x=\mu M
F=(2x^6y-3xy^5)e^{6x}+f(y)

F_y=\mu N
(2x^6-15xy^4)e^{6x}+f'(y)=(2x^6-15xy^4)e^{6x}
f'(y)=0
\implies f(y)=C

\implies F(x,y)=(2x^6y-3xy^5)e^{6x}=C
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