Question:
Solution of $\frac{dx}{yz} = \frac{dy}{zx} = \frac{dz}{xy}$ is}
Solution of $\frac{dx}{yz} = \frac{dy}{zx} = \frac{dz}{xy}$ is}
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Try to cancel out common variables between any two ratios to find an integrable pair.
Updated On: Jul 3, 2026
- $x^{2} + y^{2} = c_{1}$, $x^{2} - z^{2} = c_{2}$
- $x^{2} - y^{2} = c_{1}$, $x^{2} - z^{2} = c_{2}$
- $x^{2} - y^{2} = c_{1}$, $x^{2} + z^{2} = c_{2}$
- $x^{2} + y^{2} = c_{1}, x^{2} + z^{2} = c_{2}$
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The Correct Option is B
Solution and Explanation
Step 1: Concept
These are simultaneous differential equations solvable by the method of grouping.
Step 2: Meaning
Taking the first two fractions: $\frac{dx}{yz} = \frac{dy}{zx} \Rightarrow \frac{dx}{y} = \frac{dy}{x}$ (cancelling $z$).
Step 3: Analysis
Cross-multiplying and integrating: $x dx = y dy \Rightarrow \frac{x^{2}}{2} = \frac{y^{2}}{2} + k_{1} \Rightarrow x^{2} - y^{2} = c_{1}$. Similarly, taking the first and third fractions: $\frac{dx}{yz} = \frac{dz}{xy} \Rightarrow \frac{dx}{z} = \frac{dz}{x} \Rightarrow x dx = z dz \Rightarrow x^{2} - z^{2} = c_{2}$.
Step 4: Conclusion
The set of equations $x^{2} - y^{2} = c_{1}$ and $x^{2} - z^{2} = c_{2}$ represents the general solution. Final Answer: (B)
These are simultaneous differential equations solvable by the method of grouping.
Step 2: Meaning
Taking the first two fractions: $\frac{dx}{yz} = \frac{dy}{zx} \Rightarrow \frac{dx}{y} = \frac{dy}{x}$ (cancelling $z$).
Step 3: Analysis
Cross-multiplying and integrating: $x dx = y dy \Rightarrow \frac{x^{2}}{2} = \frac{y^{2}}{2} + k_{1} \Rightarrow x^{2} - y^{2} = c_{1}$. Similarly, taking the first and third fractions: $\frac{dx}{yz} = \frac{dz}{xy} \Rightarrow \frac{dx}{z} = \frac{dz}{x} \Rightarrow x dx = z dz \Rightarrow x^{2} - z^{2} = c_{2}$.
Step 4: Conclusion
The set of equations $x^{2} - y^{2} = c_{1}$ and $x^{2} - z^{2} = c_{2}$ represents the general solution. Final Answer: (B)
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