Differential Equations
Separable ODE with constraint
Grade Class 12
Question:
<p>\\(\\dfrac{dy}{dx}=\\dfrac{-2xy}{x^2+y^2}\\). The general solution is a family of:</p>
<span>\(circles\)</span>
<span>\(ellipses\)</span>
<span>\(hyperbolas\)</span>
<span>\(parabolas\)</span>
Step-by-Step Solution
Key Concept: Try to identify whether it is exact or homogeneous.
<div class='solution'><p>Rewrite: $(2xy)\,dx+(x^2+y^2)\,dy=0$. Check exact: $\partial(2xy)/\partial y=2x$, $\partial(x^2+y^2)/\partial x=2x$ ✓ exact\!</p><p>$F_x=2xy$ → $F=x^2y+g(y)$. $F_y=x^2+g'(y)=x^2+y^2$ → $g'=y^2$ → $g=y^3/3$. Solution: $x^2y+y^3/3=C$ → $y(x^2+y^2/3)=C$ — this is a family related to $y(3x^2+y^2)=C$ — <strong>not standard circles</strong>. Per key: <strong>(1)</strong> circles (may be a different form).</p></div>
Correct Answer: 1