164 . The solution of the differential equation ² x y d x+ ² y x d y=0 is

Step-by-step solution

The given differential equation is ² x y dx + ² y x dy = 0 . Rearranging the terms, we get ² y x dy = - ² x y dx . Separating the variables, we have ² y y dy = - ² x x dx . Integrating both sides, ² y y dy = - ² x x dx . Let u = y , then du = ² y dy . Let v = x , then dv = ² x dx . The equation becomes 1 u du = - 1 v dv . Integrating, we get |u| = - |v| + C₀ , which simplifies to |u| + |v| = C₀ . This implies |uv| = C₀ , or uv = e^ C₀ = C . Substituting back u = y and v = x , we obtain y x = C . Answer: y x=C