• anonymous
solve the DE: 2xy^3dx=(1-x^2)dy
  • Stacey Warren - Expert
Hey! We 've verified this expert answer for you, click below to unlock the details :)
At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
  • katieb
I got my questions answered at in under 10 minutes. Go to now for free help!
  • anonymous
Notice that you can manipulate this DE into a separable DE by dividing both sides by (1-x^2) and y^3 which gives you the following: \[(2x/(1-x^{2}))dx=(1/y^{3})dy\]this DE now has the separable form \[M(x)dx=N(y)dy\] Now integrate the left side of the DE with respect to x and the right side with respect to y.\[\int\limits_{}^{}(2x/(1-x^{2}))dx=\int\limits_{}^{}y^{-3}dy\]Using u=1-x^2 and performing u-substitution, the integral on the left hand side is \[-\ln(1-x^{2})+k\]and the right hand side integral is \[-1/2y^{-2}+c\] So,\[-\ln(1-x^{2}) + k = -1/2y^{-2} + c\]The c absorbs the -k when it is subtracted over to the right hand side of the equation giving you: \[-\ln(1-x^{2})=-1/2y^{-2}+c\]Now since you didn't provide an initial condition, there is no way to find an explicit solution. Only a general solution can be found because there is no initial condition to solve for the constant c. So, we simplify to get y alone. \[1/(2y^{2})=\ln(1-x^{2})+c\]\[1/2(\ln(1-x^{2})+c)=y^{2}\]\[\pm \sqrt{1/(\ln((1-x^{2})^{2})+2c)}=y\] There is no way to tell which sign on the square root is correct since no initial condition was given to determine interval of validity.

Looking for something else?

Not the answer you are looking for? Search for more explanations.