xy^3dx + e^x^2dy = 0

Question

solve this . separation of variables

Answer 1

xy³dx+e^x²dy=0,

e^x²dy=-xy³dx,

y^-3dy=-xe^-x²dx.

Let u=e^-x², du=-2xe^-x²dx, so -xe^-x²dx=du/2.

∫y^-3dy=∫-xe^-x²dx=½∫du.

-½y^-2=½u=½(e^-x²+C)=-y^-2=u=e^-x²+C, where C is integration constant.

y²=-1/(e^-x²+C)=-e^x²/(1+Ce^x²).