Question

Statement 1: The differential equation $y^{3}dy+(x+y^2)dx=0$ becomes homogeneous if we put $y^2=t$
Statement 2: All differential equation of first order and first degree of curves $f(x,y)=0$ becomes homogeneous if we put $y=tx$

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

Answer 1

The correct option is C Statement -1 is True, Statement-2 is False

$S_1:y^2=t$
$\Rightarrow 2y\frac{dy}{dx}=\frac{dt}{dx}$
$\frac{t}{2}\frac{dt}{dx}+(x+t)=0$
Which is clearly a homogeneous equation.

$S_2:$ Is not always true.
For example, $\frac{dy}{dx}=x+c$ is a differential equation of first order and first degree but it does not become homogeneous if we put $y=tx$.