Question:
The solution of $y^{4}dx + 2xy^{3}dy = \frac{ydx - xdy}{x^{3}y^{3}}$ is}
The solution of $y^{4}dx + 2xy^{3}dy = \frac{ydx - xdy}{x^{3}y^{3}}$ is}
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Look for the pattern $d(x^n y^m) = n x^{n-1} y^m dx + m x^n y^{m-1} dy$ to solve complex product differentials.
Updated On: Jul 3, 2026
- $xy^{3} + 3 \ln(\frac{y}{x}) = \text{constant}$
- $x^{3}y^{6} + 3 \ln(\frac{y}{x}) = \text{constant}$
- $x^{3}y^{6} - 3 \ln(\frac{y}{x}) = \text{constant}$
- $x^{6}y^{3} + 3 \ln(\frac{y}{x}) = \text{constant}$
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Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Step 1: Concept
This problem involves recognizing exact differentials or using substitution to simplify the expression into integrable forms.
Step 2: Meaning
Multiply the entire equation by $x^2 y^2$ to group terms. This helps in transforming the left side into a derivative of a product and the right side into a standard quotient differential.
Step 3: Analysis
Rewriting the equation: $y^3(y dx + 2x dy) = \frac{y dx - x dy}{x^3 y^3}$. Multiply by $x^2$: $x^2 y^3 (y dx + 2x dy) = \frac{x dy - y dx}{x y^3}$. Further manipulation of powers leads to the derivative of $(x y^2)^3$.
Step 4: Conclusion
Integrating the resulting forms leads to the expression $x^3 y^6$ coupled with the logarithmic term $\ln(y/x)$ as per the verified options. Final Answer: (B)
This problem involves recognizing exact differentials or using substitution to simplify the expression into integrable forms.
Step 2: Meaning
Multiply the entire equation by $x^2 y^2$ to group terms. This helps in transforming the left side into a derivative of a product and the right side into a standard quotient differential.
Step 3: Analysis
Rewriting the equation: $y^3(y dx + 2x dy) = \frac{y dx - x dy}{x^3 y^3}$. Multiply by $x^2$: $x^2 y^3 (y dx + 2x dy) = \frac{x dy - y dx}{x y^3}$. Further manipulation of powers leads to the derivative of $(x y^2)^3$.
Step 4: Conclusion
Integrating the resulting forms leads to the expression $x^3 y^6$ coupled with the logarithmic term $\ln(y/x)$ as per the verified options. Final Answer: (B)
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