Question

For any differential equation $y=f\left(x\right)$, the value of $\frac{d^{2}y}{d{x}^2}+{\left(\frac{dy}{dx}\right)}^3\frac{d^{2}x}{d{y}^2}$ is

• A. Always zero
• B. Always non-zero
• C. Equal to $2y^2$
• D. Equal to$x^2$

Answer 1

The correct option is A Always zero

$y=f\left(x\right)$

$\frac{dy}{dx}=f^{\prime }\left(x\right)$

$\frac{d^{2}y}{d{x}^2}=f^{\prime \prime }\left(x\right)$

$\frac{d^{2}y}{d{x}^2}+{\left(\frac{dy}{dx}\right)}^3\frac{d^{2}x}{d{y}^2}$

$=f^{\prime \prime }\left(x\right)+\left(f^{\prime }\right(x){)}^3\times \frac{1}{f^{\prime \prime }\left(x\right)}$

$=f^{\prime \prime }\left(x\right)+\frac{\left(f^{\prime }\right(x){)}^3}{f^{\prime \prime }\left(x\right)}$