Mathematics

Question

Verify implicit solution 2xydx+(x^2-y)dy=0; -2x^2y+y^2)=1

1 Answer

  • If [tex]-2x^2y+y^2=1[/tex], taking the derivative of both sides wrt [tex]x[/tex] gives

    [tex]-4xy-2x^2\dfrac{\mathrm dy}{\mathrm dx}+2y\dfrac{\mathrm dy}{\mathrm dx}=0[/tex]

    Solving for [tex]\dfrac{\mathrm dy}{\mathrm dx}[/tex] gives

    [tex]-4xy+(-2x^2+2y)\dfrac{\mathrm dy}{\mathrm dx}=0[/tex]
    [tex](-2x^2+2y)\dfrac{\mathrm dy}{\mathrm dx}=4xy[/tex]
    [tex](-2x^2+2y)\mathrm dy=4xy\,\mathrm dx[/tex]
    [tex](-x^2+y)\mathrm dy=2xy\,\mathrm dx[/tex]
    [tex]0=2xy\,\mathrm dx+(x^2-y)\mathrm dy[/tex]
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