Question

If $f$ and $g$ are two functions having derivative of order three for all $x$ satisfying $f\left(x\right)g\left(x\right)=C$ (constant) and $\frac{{f^{\prime }}^{\prime \prime }}{f^{\prime }}-A\frac{f^{\prime \prime }}{f}-\frac{{g^{\prime }}^{\prime \prime }}{g^{\prime }}+\frac{{3g}^{\prime \prime }}{g}=0$ Then $A$ is equal to.

Answer 1

The given equation is:

$f\left(x\right)g\left(x\right)=C$

Differentiating the equation we get,

$f\left(x\right)g^{\prime }\left(x\right)+f^{\prime }\left(x\right)g\left(x\right)=0$

$\Rightarrow f\left(x\right)g^{\prime }\left(x\right)=-f^{\prime }\left(x\right)g\left(x\right)$

Again differentiating the equation we get,

$\Rightarrow f^{\prime }\left(x\right)g^{\prime }\left(x\right)+f\left(x\right)g^{\prime \prime }\left(x\right)+f^{\prime }\left(x\right)g^{\prime }\left(x\right)+f^{\prime \prime }\left(x\right)g\left(x\right)=0$

$\Rightarrow 2f^{\prime }\left(x\right)g^{\prime }\left(x\right)+f\left(x\right)g^{\prime \prime }\left(x\right)+f^{\prime \prime }\left(x\right)g\left(x\right)=0$

Differentiating the equation for the 3rd time we get,

$\Rightarrow 2f^{\prime \prime }\left(x\right)g^{\prime }\left(x\right)+2f^{\prime }\left(x\right)g^{\prime \prime }\left(x\right)+f\left(x\right)g^{\prime \prime \prime }\left(x\right)+f^{\prime }\left(x\right)g^{\prime \prime }\left(x\right)+f^{\prime \prime }\left(x\right)g^{\prime }\left(x\right)+f^{\prime \prime \prime }\left(x\right)g\left(x\right)=0$

$\Rightarrow 3f^{\prime }\left(x\right)g^{\prime \prime }\left(x\right)+3f^{\prime \prime }\left(x\right)g^{\prime }\left(x\right)+f\left(x\right)g^{\prime \prime \prime }\left(x\right)+f^{\prime \prime \prime }\left(x\right)g\left(x\right)=0$

Dividing the whole equation by f'(x)g(x) and then using the equation obtained from 1st derivative we get,

$\Rightarrow \frac{3f^{\prime }\left(x\right)g^{\prime \prime }\left(x\right)+3f^{\prime \prime }\left(x\right)g^{\prime }\left(x\right)+f\left(x\right)g^{\prime \prime \prime }\left(x\right)+f^{\prime \prime \prime }\left(x\right)g\left(x\right)}{f^{\prime }\left(x\right)g\left(x\right)}=0$

$\Rightarrow \frac{f^{\prime \prime \prime }\left(x\right)g\left(x\right)+3f^{\prime }\left(x\right)g^{\prime \prime }\left(x\right)}{f^{\prime }\left(x\right)g\left(x\right)}-\frac{3f^{\prime \prime }\left(x\right)g^{\prime }\left(x\right)+f\left(x\right)g^{\prime \prime \prime }\left(x\right)}{f\left(x\right)g^{\prime }\left(x\right)}=0$

$\Rightarrow \frac{f^{\prime \prime \prime }}{f^{\prime }}+3\frac{g^{\prime \prime }}{g}-\frac{g^{\prime \prime \prime }}{g^{\prime }}-3\frac{f^{\prime \prime }}{f}=0$

$\Rightarrow A=3$ ....Answer