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Consider the differential equation 2 cos (y2)dx - xy sin (y2)dy = 0
- a)ex is an integrating factor
- b)e-x is an integrating factor
- c)3x is an integrating factor
- d)x3 is an integrating factor
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
Consider the differential equation 2 cos (y2)dx - xy sin (y2)dy = 0a)e...
we have 2cos (y2)dx - xy sin (y2) dy = 0 Multiplying by x3
and
Since,
So, x3 is integrating factor.
and
Since,
So, x3 is integrating factor.
Most Upvoted Answer
Consider the differential equation 2 cos (y2)dx - xy sin (y2)dy = 0a)e...
Given: The differential equation 2 cos (y^2)dx - xy sin (y^2)dy = 0
To solve this differential equation, we can use the method of integrating factors. An integrating factor is a function that we can multiply to the entire equation to make it exact.
Step 1: Check if the equation is exact:
An equation of the form M(x, y)dx + N(x, y)dy = 0 is exact if and only if ∂M/∂y = ∂N/∂x.
Let's find the partial derivatives:
∂M/∂y = -4ycos(y^2)
∂N/∂x = -ysin(y^2)
Since ∂M/∂y is not equal to ∂N/∂x, the given differential equation is not exact.
Step 2: Find the integrating factor:
To find the integrating factor, we can use the formula: integrating factor = e^(∫(∂N/∂x - ∂M/∂y)/N dx).
In this case, ∂M/∂y - ∂N/∂x = -4ycos(y^2) - (-ysin(y^2)) = -4ycos(y^2) + ysin(y^2).
We need to find the integral of (-4ycos(y^2) + ysin(y^2))/(-xy sin(y^2)) dx.
Simplifying the expression, we get (∫(4cos(y^2)/y - sin(y^2))/x dx.
The integral (∫4cos(y^2)/y dx) can be solved using the substitution method, and the integral (∫sin(y^2)/x dx) can be solved using the logarithmic method.
After finding the integral, we get the integrating factor as x^4/y^4.
Step 3: Multiply the integrating factor to the equation:
Multiplying the integrating factor x^4/y^4 to the given differential equation, we get:
(2x^3/y^4)cos(y^2)dx - (x^5/y^3)sin(y^2)dy = 0
This equation is exact, and we can solve it by finding the potential function or by using other methods.
Therefore, the correct answer is option D) x^3 is an integrating factor.
To solve this differential equation, we can use the method of integrating factors. An integrating factor is a function that we can multiply to the entire equation to make it exact.
Step 1: Check if the equation is exact:
An equation of the form M(x, y)dx + N(x, y)dy = 0 is exact if and only if ∂M/∂y = ∂N/∂x.
Let's find the partial derivatives:
∂M/∂y = -4ycos(y^2)
∂N/∂x = -ysin(y^2)
Since ∂M/∂y is not equal to ∂N/∂x, the given differential equation is not exact.
Step 2: Find the integrating factor:
To find the integrating factor, we can use the formula: integrating factor = e^(∫(∂N/∂x - ∂M/∂y)/N dx).
In this case, ∂M/∂y - ∂N/∂x = -4ycos(y^2) - (-ysin(y^2)) = -4ycos(y^2) + ysin(y^2).
We need to find the integral of (-4ycos(y^2) + ysin(y^2))/(-xy sin(y^2)) dx.
Simplifying the expression, we get (∫(4cos(y^2)/y - sin(y^2))/x dx.
The integral (∫4cos(y^2)/y dx) can be solved using the substitution method, and the integral (∫sin(y^2)/x dx) can be solved using the logarithmic method.
After finding the integral, we get the integrating factor as x^4/y^4.
Step 3: Multiply the integrating factor to the equation:
Multiplying the integrating factor x^4/y^4 to the given differential equation, we get:
(2x^3/y^4)cos(y^2)dx - (x^5/y^3)sin(y^2)dy = 0
This equation is exact, and we can solve it by finding the potential function or by using other methods.
Therefore, the correct answer is option D) x^3 is an integrating factor.
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Consider the differential equation 2 cos (y2)dx - xy sin (y2)dy = 0a)ex is an integrating factorb)e-x is an integrating factorc)3x is an integrating factord)x3 is an integrating factorCorrect answer is option 'D'. Can you explain this answer?
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