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If P(y) and Q(x) are arbitrary functions of y and a respectively, then the differential equation P(y)dx+ Q(x)dy = 0
- a)Is never exact
- b)Is exact if P(y) = ey and Q(x) = ex
- c)Is exact only if P(y) = Ay + B and Q(x) = Ax + C, A, B, C being constants
- d)Is always exact
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
If P(y) and Q(x) are arbitrary functions of y and a respectively, then...
Proof : The given differential equation is
P(y)dx+Q(x)dy = 0 ...(i)
Comparing (i) with
Mdx + Ndy= 0 ...(ii)
M = P(y) and N = Q(x)
Differential equation (i) will be exact if and only if
...(iii)
P(y)dx+Q(x)dy = 0 ...(i)
Comparing (i) with
Mdx + Ndy= 0 ...(ii)
M = P(y) and N = Q(x)
Differential equation (i) will be exact if and only if
...(iii)Most Upvoted Answer
If P(y) and Q(x) are arbitrary functions of y and a respectively, then...
Proof : The given differential equation is
P(y)dx+Q(x)dy = 0 ...(i)
Comparing (i) with
Mdx + Ndy= 0 ...(ii)
M = P(y) and N = Q(x)
Differential equation (i) will be exact if and only if
...(iii)
P(y)dx+Q(x)dy = 0 ...(i)
Comparing (i) with
Mdx + Ndy= 0 ...(ii)
M = P(y) and N = Q(x)
Differential equation (i) will be exact if and only if
...(iii)Free Test
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Community Answer
If P(y) and Q(x) are arbitrary functions of y and a respectively, then...
Understanding Exact Differential Equations
In the context of differential equations, the equation P(y)dx + Q(x)dy = 0 is called an exact differential equation if it satisfies a certain condition.
Condition for Exactness
An equation of the form M(x, y)dx + N(x, y)dy = 0 is exact if the following condition holds:
∂M/∂y = ∂N/∂x
For our equation:
- M = P(y) (depends only on y)
- N = Q(x) (depends only on x)
Here, M does not depend on x, and N does not depend on y. Therefore, the derivatives ∂M/∂y and ∂N/∂x are:
- ∂M/∂y = P'(y)
- ∂N/∂x = Q'(x)
Since these two derivatives do not relate to each other (one depends on y and the other on x), the condition for exactness fails.
Analysis of the Options
- Option A: "Is never exact" - This is incorrect; while it may not generally be exact, specific forms of P and Q can make it exact.
- Option B: "Is exact if P(y) = ey and Q(x) = e^x" - This is a special case and does not represent the general scenario.
- Option C: "Is exact only if P(y) = Ay + B and Q(x) = Ax + C, A, B, C being constants" - This is correct. Here, the derivatives of P and Q will yield constant relationships, satisfying the exactness condition.
- Option D: "Is always exact" - This is incorrect as it overlooks the need for specific forms of P and Q.
Conclusion
In summary, the equation P(y)dx + Q(x)dy = 0 is exact under specific linear conditions for P and Q, validating option C as the correct answer.
In the context of differential equations, the equation P(y)dx + Q(x)dy = 0 is called an exact differential equation if it satisfies a certain condition.
Condition for Exactness
An equation of the form M(x, y)dx + N(x, y)dy = 0 is exact if the following condition holds:
∂M/∂y = ∂N/∂x
For our equation:
- M = P(y) (depends only on y)
- N = Q(x) (depends only on x)
Here, M does not depend on x, and N does not depend on y. Therefore, the derivatives ∂M/∂y and ∂N/∂x are:
- ∂M/∂y = P'(y)
- ∂N/∂x = Q'(x)
Since these two derivatives do not relate to each other (one depends on y and the other on x), the condition for exactness fails.
Analysis of the Options
- Option A: "Is never exact" - This is incorrect; while it may not generally be exact, specific forms of P and Q can make it exact.
- Option B: "Is exact if P(y) = ey and Q(x) = e^x" - This is a special case and does not represent the general scenario.
- Option C: "Is exact only if P(y) = Ay + B and Q(x) = Ax + C, A, B, C being constants" - This is correct. Here, the derivatives of P and Q will yield constant relationships, satisfying the exactness condition.
- Option D: "Is always exact" - This is incorrect as it overlooks the need for specific forms of P and Q.
Conclusion
In summary, the equation P(y)dx + Q(x)dy = 0 is exact under specific linear conditions for P and Q, validating option C as the correct answer.
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If P(y) and Q(x) are arbitrary functions of y and a respectively, then the differential equation P(y)dx+ Q(x)dy = 0a)Is never exactb)Is exact if P(y) = ey and Q(x) = exc)Is exact only if P(y) = Ay + B and Q(x) = Ax + C, A, B, C being constantsd)Is always exactCorrect answer is option 'C'. Can you explain this answer?
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