Question

The general solution of $x^{4}p^{2} = y + px$, $p = \frac{dy}{dx}$ is}

• A. $y = cx + c^{2}$
• B. $xy = c^{2}x + c$
• C. $x = c^{2}y + c$
• D. $xy = c^{2}y + c$

Hint:

When you see high powers of $x$ and $p$ together, consider substitutions like $1/x$ or $x^2$ to find a standard form.

Answer 1

The Correct Option is B

Step 1: Concept
This equation can be transformed into Clairaut's form using a substitution such as $u = 1/x$.

Step 2: Meaning
Let $x = 1/u$. Then $dx = -1/u^2 du$. The derivative $p = dy/dx$ transforms accordingly.

Step 3: Analysis
Substituting and simplifying the expression $x^4 p^2 = y + px$ leads to a linear relationship in the transformed variables that matches the structure of a modified Clairaut's equation.

Step 4: Conclusion
Applying the general solution method for the transformed equation and returning to original variables $x$ and $y$ yields $xy = c^{2}x + c$ as the general solution. Final Answer: (B)