Question

Show that the solution y(x) to the equation sqrt(a2-y2)- alna(a+sqrt(a2-y2))+aln(y)+c=0. Is the solution of the differential equation dy/dx= -y/sqrt(a2-y2)

Answer 1

$\intop \frac{\sqrt{a^2-y^2}}{y}dy=-x+c_1$

$\intop \frac{\sqrt{a^2-y^2}}{y}dy=\sqrt{a^2-y^2}-aln(\frac{\sqrt{a^2-y^2}+a}{y})+c_2$

$\sqrt{a^2-y^2}-aln(\frac{\sqrt{a^2-y^2}+a}{y})+c_2=-x+c_1$

$\sqrt{a^2-y^2}-aln({\sqrt{a^2-y^2}+a})+alny+c=-x$