Question

Statement-1 : The differential equation of the form $yf(xy)dx+x\varphi (xy)dy=0$ can be converted to homogeneous forms by substitution $xy=v$.
Statement-2 : All differential equation of first order and first degree become homogeneous, if we put $y=vx$.

• A. Statement-1 is true, Statement-2 is true, Statement-2 is a correct explanation for Statement -1
• B. Statement -1 is True, Statement -2 is True ; Statement- 2 is NOT a correct explanation for Statement - 1
• C. Statement -1 is False, Statement -2 is True
• D. Statement - 1 is True, Statement- 2 is False

Answer 1

Answer: (C)

$\because xy=v$
$\therefore x\frac{dy}{dx}+y=\frac{dv}{dx}$
Then, the given equation reduces to
$\frac{v}{x}f\left(v\right)+x\varphi \left(v\right)\left(\frac{1}{x}\left(\frac{dv}{dx}-y\right)\right)=0$
$\Rightarrow \frac{v}{x}f\left(v\right)+\varphi \left(v\right)\frac{dv}{dx}-y\varphi \left(v\right)=0$
$\Rightarrow \left\{\frac{v\left(f\left(v\right)-\varphi \left(v\right)\right)}{x}\right\}+\varphi \left(v\right)\frac{dv}{dx}=0$
$\Rightarrow \frac{dx}{x}+\frac{\varphi \left(v\right)dv}{vf\left(v\right)-\varphi \left(v\right)}=0$
Which is variable seperable form.