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An integrating factor for (cos y sin 2x) dx + (cos2 y - cos2 x)dy = 0 is
- a)1/(sec2 y + sec y tan y)
- b)tan2 y + sec y tan y
- c)sec2 y + sec y tan y
- d)1/(tan2y + sec y tan y)
Correct answer is option 'C'. Can you explain this answer?
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An integrating factor for (cos y sin 2x)dx + (cos2 y - cos2 x)dy = 0 i...
Given equation:
(cos y sin 2x)dx + (cos2 y - cos2 x)dy = 0
Step 1: Identify the form of the equation
The given equation is not in the standard form for exact differential equations, which is M(x, y)dx + N(x, y)dy = 0. To transform it into this form, we need to multiply an integrating factor.
Step 2: Find the integrating factor
To find the integrating factor, we need to check if the equation satisfies the exactness condition, which states that ∂M/∂y = ∂N/∂x.
In this case,
∂(cos y sin 2x)/∂y = cos 2x
∂(cos2 y - cos2 x)/∂x = 2cos2 x
Since cos 2x and 2cos2 x are not equal, the equation is not exact. Therefore, we need to find an integrating factor.
Step 3: Find the integrating factor
The integrating factor (IF) for a first-order linear differential equation of the form M(x, y)dx + N(x, y)dy = 0 can be found using the formula:
IF = e^(∫(∂M/∂y - ∂N/∂x)/N dx)
In this case, we have:
∂M/∂y - ∂N/∂x = cos 2x - 2cos2 x
IF = e^(∫(cos 2x - 2cos2 x)/(cos2 y - cos2 x) dx)
Step 4: Simplify the integral
To simplify the integral, we can use the trigonometric identity cos 2x = 2cos^2 x - 1.
IF = e^(∫(2cos^2 x - 2cos2 x)/(cos2 y - cos2 x) dx)
IF = e^(∫(2cos^2 x - 2(2cos^2 x - 1))/(cos2 y - cos2 x) dx)
IF = e^(∫(2 - 4cos^2 x)/(cos2 y - cos2 x) dx)
IF = e^(∫(2(1 - 2cos^2 x))/(cos2 y - cos2 x) dx)
Step 5: Simplify the integral further
We can simplify the integral by using the trigonometric identity sin^2 x = 1 - cos^2 x.
IF = e^(∫(2(1 - 2cos^2 x))/(cos2 y - cos2 x) dx)
IF = e^(∫(2(1 - 2(1 - sin^2 x)))/(cos2 y - cos2 x) dx)
IF = e^(∫(2(1 - 2 + 2sin^2 x))/(cos2 y - cos2 x) dx)
IF = e^(∫(4sin^2 x - 2)/(cos2 y - cos2 x) dx)
Step 6: Evaluate the integral
Let's integrate the expression inside the exponential:
∫(4
(cos y sin 2x)dx + (cos2 y - cos2 x)dy = 0
Step 1: Identify the form of the equation
The given equation is not in the standard form for exact differential equations, which is M(x, y)dx + N(x, y)dy = 0. To transform it into this form, we need to multiply an integrating factor.
Step 2: Find the integrating factor
To find the integrating factor, we need to check if the equation satisfies the exactness condition, which states that ∂M/∂y = ∂N/∂x.
In this case,
∂(cos y sin 2x)/∂y = cos 2x
∂(cos2 y - cos2 x)/∂x = 2cos2 x
Since cos 2x and 2cos2 x are not equal, the equation is not exact. Therefore, we need to find an integrating factor.
Step 3: Find the integrating factor
The integrating factor (IF) for a first-order linear differential equation of the form M(x, y)dx + N(x, y)dy = 0 can be found using the formula:
IF = e^(∫(∂M/∂y - ∂N/∂x)/N dx)
In this case, we have:
∂M/∂y - ∂N/∂x = cos 2x - 2cos2 x
IF = e^(∫(cos 2x - 2cos2 x)/(cos2 y - cos2 x) dx)
Step 4: Simplify the integral
To simplify the integral, we can use the trigonometric identity cos 2x = 2cos^2 x - 1.
IF = e^(∫(2cos^2 x - 2cos2 x)/(cos2 y - cos2 x) dx)
IF = e^(∫(2cos^2 x - 2(2cos^2 x - 1))/(cos2 y - cos2 x) dx)
IF = e^(∫(2 - 4cos^2 x)/(cos2 y - cos2 x) dx)
IF = e^(∫(2(1 - 2cos^2 x))/(cos2 y - cos2 x) dx)
Step 5: Simplify the integral further
We can simplify the integral by using the trigonometric identity sin^2 x = 1 - cos^2 x.
IF = e^(∫(2(1 - 2cos^2 x))/(cos2 y - cos2 x) dx)
IF = e^(∫(2(1 - 2(1 - sin^2 x)))/(cos2 y - cos2 x) dx)
IF = e^(∫(2(1 - 2 + 2sin^2 x))/(cos2 y - cos2 x) dx)
IF = e^(∫(4sin^2 x - 2)/(cos2 y - cos2 x) dx)
Step 6: Evaluate the integral
Let's integrate the expression inside the exponential:
∫(4
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An integrating factor for (cos y sin 2x)dx + (cos2 y - cos2 x)dy = 0 isa)1/(sec2 y + sec y tan y)b)tan2 y + sec y tan yc)sec2 y + sec y tan yd)1/(tan2y + sec y tan y)Correct answer is option 'C'. Can you explain this answer?
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An integrating factor for (cos y sin 2x)dx + (cos2 y - cos2 x)dy = 0 isa)1/(sec2 y + sec y tan y)b)tan2 y + sec y tan yc)sec2 y + sec y tan yd)1/(tan2y + sec y tan y)Correct answer is option 'C'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about An integrating factor for (cos y sin 2x)dx + (cos2 y - cos2 x)dy = 0 isa)1/(sec2 y + sec y tan y)b)tan2 y + sec y tan yc)sec2 y + sec y tan yd)1/(tan2y + sec y tan y)Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for An integrating factor for (cos y sin 2x)dx + (cos2 y - cos2 x)dy = 0 isa)1/(sec2 y + sec y tan y)b)tan2 y + sec y tan yc)sec2 y + sec y tan yd)1/(tan2y + sec y tan y)Correct answer is option 'C'. Can you explain this answer?.
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