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The integrating factor of equation y log y dx + (x – log y) dy = 0 is
- a)log x
- b)log y
- c)log (log x)
- d)log (log y)
Correct answer is option 'B'. Can you explain this answer?
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The integrating factor of equation y log y dx + (x – log y) dy =...
Integrating Factor for the Given Differential Equation
The given differential equation is in the form of y log y dx + (x – log y) dy = 0. To find the integrating factor for this differential equation, we can rewrite it in the form of Mdx + Ndy = 0, where M = y log y and N = x – log y.
Rewriting the Differential Equation
The given differential equation can be rewritten as y log y dx + x dy – log y dy = 0.
Now, we have M = y log y and N = x – log y.
Finding the Integrating Factor
The integrating factor (denoted by μ) for the differential equation in the form Mdx + Ndy = 0 is given by the formula:
μ = e^(∫(N-M)/M dx)
In this case, (N-M)/M = (x - log y - y log y)/ (y log y) = (x/y - log y - log y) = x/y - 2log y.
So, the integrating factor μ = e^(∫(x/y - 2log y) dx) = e^(x/y - 2log y) = e^(x/y) / y^2.
Therefore, the integrating factor for the given differential equation y log y dx + (x – log y) dy = 0 is log y (option B).
The given differential equation is in the form of y log y dx + (x – log y) dy = 0. To find the integrating factor for this differential equation, we can rewrite it in the form of Mdx + Ndy = 0, where M = y log y and N = x – log y.
Rewriting the Differential Equation
The given differential equation can be rewritten as y log y dx + x dy – log y dy = 0.
Now, we have M = y log y and N = x – log y.
Finding the Integrating Factor
The integrating factor (denoted by μ) for the differential equation in the form Mdx + Ndy = 0 is given by the formula:
μ = e^(∫(N-M)/M dx)
In this case, (N-M)/M = (x - log y - y log y)/ (y log y) = (x/y - log y - log y) = x/y - 2log y.
So, the integrating factor μ = e^(∫(x/y - 2log y) dx) = e^(x/y - 2log y) = e^(x/y) / y^2.
Therefore, the integrating factor for the given differential equation y log y dx + (x – log y) dy = 0 is log y (option B).
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The integrating factor of equation y log y dx + (x – log y) dy = 0 isa)log xb)log yc)log (log x)d)log (log y)Correct answer is option 'B'. Can you explain this answer?
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