Question

The solution of the differential equation \(x^2\,dx+y^2\,dy=0\) is

• A. \(x^3+y^3=k\)
• B. \(x^2+y^2=k\)
• C. \(x^2-y^2=k\)
• D. \(x^2-y^2=k\)

Hint:

Form \(M(x)dx+N(y)dy=0\) \(\Rightarrow\) integrate terms independently.

Answer 1

The Correct Option is A

Idea. This is already separated: an \(x\)-part with \(dx\) plus a \(y\)-part with \(dy\). Integrate each side straight away.
\[ \int x^2\,dx+\int y^2\,dy=C $\Rightarrow$ \frac{x^3}{3}+\frac{y^3}{3}=C $\Rightarrow$ x^3+y^3=k. \]