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If P(x) and Q(y) are arbitrary functions of x and y respectively, then the differential equation P(x)dx + Q(y)dy = 0
- a)May or may not be exact
- b)Is never exact
- c)Is always exact
- d)Is exact only when P(x) = x and Q(y) = y
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
If P(x) and Q(y) are arbitrary functions of x and y respectively, then...
Proof : The given differential equation is
P(x)dx+Q(y)dy = 0 ...(i)
Comparing it with
Mdx + Ndy = 0 ...(ii)
∴ M = P(x) and N = Q(y)
is always satisfied.
⇒ D.E. (i) is always exact.
P(x)dx+Q(y)dy = 0 ...(i)
Comparing it with
Mdx + Ndy = 0 ...(ii)
∴ M = P(x) and N = Q(y)
is always satisfied.
⇒ D.E. (i) is always exact.
Most Upvoted Answer
If P(x) and Q(y) are arbitrary functions of x and y respectively, then...
The given differential equation is P(x)dx + Q(y)dy = 0, where P(x) and Q(y) are arbitrary functions of x and y respectively. We are asked to determine if this differential equation is exact or not.
To determine if a differential equation is exact, we need to check if it satisfies the condition:
∂(M)/∂(y) = ∂(N)/∂(x)
where M = P(x) and N = Q(y).
Let's calculate the partial derivatives:
∂(M)/∂(y) = 0 (since M = P(x) does not involve y)
∂(N)/∂(x) = 0 (since N = Q(y) does not involve x)
Since the partial derivatives of M and N with respect to their respective variables are both zero, the given differential equation satisfies the condition for exactness:
∂(M)/∂(y) = ∂(N)/∂(x)
Thus, we can conclude that the given differential equation is exact.
Explanation:
The given differential equation has the form M(x, y)dx + N(x, y)dy = 0, where M = P(x) and N = Q(y). For a differential equation to be exact, it must satisfy the condition ∂(M)/∂(y) = ∂(N)/∂(x).
In this case, since M = P(x) does not involve y and N = Q(y) does not involve x, the partial derivatives with respect to y and x, respectively, are zero. Therefore, ∂(M)/∂(y) = ∂(N)/∂(x) = 0.
Since the partial derivatives are equal to zero, the given differential equation satisfies the condition for exactness. Hence, option C, "Is always exact," is the correct answer.
Note: The fact that P(x) = x and Q(y) = y are not necessary conditions for the exactness of the differential equation. The equation is exact regardless of the specific forms of P(x) and Q(y).
To determine if a differential equation is exact, we need to check if it satisfies the condition:
∂(M)/∂(y) = ∂(N)/∂(x)
where M = P(x) and N = Q(y).
Let's calculate the partial derivatives:
∂(M)/∂(y) = 0 (since M = P(x) does not involve y)
∂(N)/∂(x) = 0 (since N = Q(y) does not involve x)
Since the partial derivatives of M and N with respect to their respective variables are both zero, the given differential equation satisfies the condition for exactness:
∂(M)/∂(y) = ∂(N)/∂(x)
Thus, we can conclude that the given differential equation is exact.
Explanation:
The given differential equation has the form M(x, y)dx + N(x, y)dy = 0, where M = P(x) and N = Q(y). For a differential equation to be exact, it must satisfy the condition ∂(M)/∂(y) = ∂(N)/∂(x).
In this case, since M = P(x) does not involve y and N = Q(y) does not involve x, the partial derivatives with respect to y and x, respectively, are zero. Therefore, ∂(M)/∂(y) = ∂(N)/∂(x) = 0.
Since the partial derivatives are equal to zero, the given differential equation satisfies the condition for exactness. Hence, option C, "Is always exact," is the correct answer.
Note: The fact that P(x) = x and Q(y) = y are not necessary conditions for the exactness of the differential equation. The equation is exact regardless of the specific forms of P(x) and Q(y).
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Community Answer
If P(x) and Q(y) are arbitrary functions of x and y respectively, then...
Proof : The given differential equation is
P(x)dx+Q(y)dy = 0 ...(i)
Comparing it with
Mdx + Ndy = 0 ...(ii)
∴ M = P(x) and N = Q(y)
is always satisfied.
⇒ D.E. (i) is always exact.
P(x)dx+Q(y)dy = 0 ...(i)
Comparing it with
Mdx + Ndy = 0 ...(ii)
∴ M = P(x) and N = Q(y)
is always satisfied.
⇒ D.E. (i) is always exact.
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If P(x) and Q(y) are arbitrary functions of x and y respectively, then the differential equation P(x)dx +Q(y)dy = 0a)May or may not be exactb)Is never exactc)Is always exactd)Is exact only when P(x) = xand Q(y) = yCorrect answer is option 'C'. Can you explain this answer?
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