If $\frac{dy}{dx} + y\frac{dx}{dy} = x.y(-2) = 1$, then :
Step-by-Step Solution
Key Concept: Recognize this as Clairaut's equation in the form y = xp - p² where p = dy/dx. Find both the general solution y = cx - c² and the singular solution (envelope) by eliminating c from y = cx - c² and ∂/∂c(y - cx + c²) = 0, which gives c = x/2, yielding the singular solution x² - 4y = 0. Apply the initial condition y(-2) = 1 to verify which solutions satisfy it.
This is Clairaut's equation $y = xp - p^2$ where $p = rac{dy}{dx}$. The general solution is $y = cx - c^2$. Setting the discriminant condition for singular solution: the envelope is found from $ rac{\partial}{\partial c}(y - cx + c^2) = 0 \Rightarrow -x + 2c = 0 \Rightarrow c = rac{x}{2}$. Substituting back: $y = x \cdot rac{x}{2} - ( rac{x}{2})^2 = rac{x^2}{4}$.
Correct Answer: 3,4