Question:
The differential equation $\frac{dy}{dx} = \frac{-(x + x^{8} + py^{2})}{y^{8} - y + qxy}$ is exact if}
The differential equation $\frac{dy}{dx} = \frac{-(x + x^{8} + py^{2})}{y^{8} - y + qxy}$ is exact if}
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Exactness is the "cross-derivative" check. If $M_y = N_x$, the path integral is independent of the path.
Updated On: Jul 3, 2026
- $2p = q$
- $p = q$
- $p \neq 2q$
- $p = 1, q = 3$
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The Correct Option is A
Solution and Explanation
Step 1: Concept
A differential equation $M dx + N dy = 0$ is exact if $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$.
Step 2: Meaning
Rewrite the given equation: $(x + x^{8} + py^{2}) dx + (y^{8} - y + qxy) dy = 0$. Here $M = x + x^{8} + py^{2}$ and $N = y^{8} - y + qxy$.
Step 3: Analysis
Calculate partial derivatives: $\frac{\partial M}{\partial y} = 2py$ and $\frac{\partial N}{\partial x} = qy$.
Step 4: Conclusion
For the equation to be exact, we set $2py = qy$. Solving for the constants gives $2p = q$. Final Answer: (A)
A differential equation $M dx + N dy = 0$ is exact if $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$.
Step 2: Meaning
Rewrite the given equation: $(x + x^{8} + py^{2}) dx + (y^{8} - y + qxy) dy = 0$. Here $M = x + x^{8} + py^{2}$ and $N = y^{8} - y + qxy$.
Step 3: Analysis
Calculate partial derivatives: $\frac{\partial M}{\partial y} = 2py$ and $\frac{\partial N}{\partial x} = qy$.
Step 4: Conclusion
For the equation to be exact, we set $2py = qy$. Solving for the constants gives $2p = q$. Final Answer: (A)
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