Q.If $ax^2 + 2hxy + by^2 = 0$, then prove that $\dfrac{d^2y}{dx^2} = 0$.

Because $ax^2+2hxy+by^2=0$ is homogeneous, $y/x$ is a constant, so $y=mx$ with $\dfrac{dy}{dx}=m$ constant, and therefore $\dfrac{d^2y}{dx^2}=0$.

Method — reduce to $y=mx$. The given relation is homogeneous of degree $2$ in $x$ and $y$. Put $y=vx$, where $v=\dfrac{y}{x}$. Substituting:

$ax^2+2hx(vx)+b(vx)^2=0\ \Rightarrow\ x^2\left(a+2hv+bv^2\right)=0$.

For $x\neq 0$ this gives $bv^2+2hv+a=0$, an equation with no $x$ in it. Hence $v$ takes only constant values (the two roots $m_1,m_2$); that is, $\dfrac{y}{x}=m$ (constant), so

$y=mx$.

Differentiating: $\dfrac{dy}{dx}=m$, a constant. Differentiating once more:

$\dfrac{d^2y}{dx^2}=\dfrac{d}{dx}(m)=0$.

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