Exact De
The differential equation M dx + N dy = 0 is exact if
Select the correct option:
🔒 Solution Hidden from View
Submit your answer to unlock the detailed step-by-step solution.
More differential equations Practice Questions
A cup of coffee cools according to Newton's law so that \frac{dT}{dt} = -k(T - T_s); identify the co...
A cup of coffee cools according to Newton's law so that \frac{dT}{dt} = -k(T - T_s); identify the co...
Solve the repeated-root equation \frac{d^2y}{dx^2} - 6\frac{dy}{dx} + 9y = 0 and state the general s...
Solve the repeated-root equation \frac{d^2y}{dx^2} - 6\frac{dy}{dx} + 9y = 0 and state the general s...
The length of the subtangent at any point of a curve is constant and equal to a, leading to a differ...
The length of the subtangent at any point of a curve is constant and equal to a, leading to a differ...
Solve the differential equation \frac{dy}{dx} = e^{x-y} and express the general solution relating th...
Solve the differential equation \frac{dy}{dx} = e^{x-y} and express the general solution relating th...
The family of curves y = Cx^2 has orthogonal trajectories that we must determine by replacing the sl...
The family of curves y = Cx^2 has orthogonal trajectories that we must determine by replacing the sl...
Find the general solution of the second-order constant-coefficient equation \frac{d^2y}{dx^2} - 5\fr...
Find the general solution of the second-order constant-coefficient equation \frac{d^2y}{dx^2} - 5\fr...
A population grows so that its rate of change is proportional to its current size, modeled by \frac{...
A population grows so that its rate of change is proportional to its current size, modeled by \frac{...
Consider the equation x\frac{dy}{dx} + 2y = x^2 with x positive, and identify the general solution a...
Consider the equation x\frac{dy}{dx} + 2y = x^2 with x positive, and identify the general solution a...
At every point on a curve the slope equals the ratio of the y-coordinate to the x-coordinate, and we...
At every point on a curve the slope equals the ratio of the y-coordinate to the x-coordinate, and we...
Determine the general solution of the equation \frac{d^2y}{dx^2} + 4y = 0 whose auxiliary equation p...
Determine the general solution of the equation \frac{d^2y}{dx^2} + 4y = 0 whose auxiliary equation p...
Examine the equation \sqrt{1 + \left(\frac{dy}{dx}\right)^2} = \frac{d^2y}{dx^2} and determine the d...
Examine the equation \sqrt{1 + \left(\frac{dy}{dx}\right)^2} = \frac{d^2y}{dx^2} and determine the d...
Solve the homogeneous equation \frac{dy}{dx} = \frac{y^2}{xy - x^2} by substituting y equal to v tim...
Solve the homogeneous equation \frac{dy}{dx} = \frac{y^2}{xy - x^2} by substituting y equal to v tim...
About This Question
- Subject
- mathematics
- Chapter
- differential equations
- Topic
- exact de
- Difficulty
- Medium
- Year
- 2025
Solution
Correct Answer:
∂M/∂y = ∂N/∂x
- Definition: An equation is 'exact' if it is the total differential of some function u(x,y), i.e., du=Mdx+Ndy.
- Symmetry Condition: For du=fxdx+fydy, Clairaut's theorem states that fxy=fyx for continuous functions.
- Translation: Since M=∂x∂u and N=∂y∂u, we must have:
- ∂y∂M=∂x∂y∂2u and ∂x∂N=∂y∂x∂2u.
- Conclusion: The necessary and sufficient condition is ∂y∂M=∂x∂N.
This medium difficulty mathematics question is from the chapter differential equations, covering the topic of exact de. It appeared in the 2025 exam.
Looking for more practice? Explore all mathematics questions or browse differential equations questions on RankGuru.