Question

Statement-1 : The differential equation of the form $yf (xy) dx + x\phi (xy) dy = 0$ 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)

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