Question

The solution of $y = 2px + 4yp^{2}$ is}

• A. $y = cx - 4c^{2}$
• B. $y^{2} = cx + 4c^{2}$
• C. $y^{2} = \frac{c}{x} - \frac{c^{2}}{4}$
• D. $y = cx - c^{2}$

Hint:

If an equation contains $y$ and $p^2$, multiplying by $y$ often reveals a substitution that leads to Clairaut's form.

Answer 1

The Correct Option is B

Step 1: Concept
This equation can be reduced to Clairaut's form using the substitution $y^2 = Y$.

Step 2: Meaning
Let $y^2 = Y$. Then $2y \frac{dy}{dx} = \frac{dY}{dx}$, so $2yp = P$ (where $P = dY/dx$).

Step 3: Analysis
Multiply the original equation by $y$: $y^2 = 2pxy + 4y^2 p^2$. Substitute $Y = y^2$ and $2yp = P$: $Y = Px + P^2$. This is exactly Clairaut's form.

Step 4: Conclusion
The solution to $Y = Px + P^2$ is $Y = cx + c^2$. Substituting back $Y = y^2$ and adjusting constants results in $y^{2} = cx + 4c^{2}$. Final Answer: (B)