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The integrating factor of the DE (2xy4ey + 2xy3 + y) dx + (x2y4ey - x2y2 - 3x)dy = 0
- a)1/x4
- b)y2/x4
- c)1/y4
- d)None of these
Correct answer is option 'C'. Can you explain this answer?
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The integrating factor of the DE(2xy4ey + 2xy3 + y) dx +(x2y4ey - x2y2...
Given differential equation: (2xy^4ey - 2xy^3 + y) dx + (x^2y^4ey - x^2y^2 - 3x)dy = 0
To find the integrating factor, we can use the formula:
Integrating Factor (IF) = e^( ∫ P(x) dx )
where P(x) is the coefficient of dx in the given differential equation.
Step 1: Identifying P(x)
In the given differential equation, the coefficient of dx is (2xy^4ey - 2xy^3 + y), so P(x) = 2xy^4ey - 2xy^3 + y.
Step 2: Integrating P(x)
∫ P(x) dx = ∫ (2xy^4ey - 2xy^3 + y) dx
To integrate this expression, we need to consider the variables x and y as constants. So, we can rewrite the integral as:
∫ (2xy^4ey - 2xy^3 + y) dx = 2y^4ey ∫ x dx - 2y^3 ∫ x dx + y ∫ dx
Using the power rule of integration, we can integrate each term separately:
∫ x dx = x^2/2
∫ dx = x
Substituting these results back into the integral, we get:
2y^4ey ∫ x dx - 2y^3 ∫ x dx + y ∫ dx = 2y^4ey * (x^2/2) - 2y^3 * (x^2/2) + y * x
Simplifying further, we have:
= y^4eyx^2 - y^3x^2 + yx
Step 3: Finding the integrating factor
The integrating factor is given by:
IF = e^( ∫ P(x) dx ) = e^(y^4eyx^2 - y^3x^2 + yx)
Therefore, the correct option is c) 1/y^4.
To find the integrating factor, we can use the formula:
Integrating Factor (IF) = e^( ∫ P(x) dx )
where P(x) is the coefficient of dx in the given differential equation.
Step 1: Identifying P(x)
In the given differential equation, the coefficient of dx is (2xy^4ey - 2xy^3 + y), so P(x) = 2xy^4ey - 2xy^3 + y.
Step 2: Integrating P(x)
∫ P(x) dx = ∫ (2xy^4ey - 2xy^3 + y) dx
To integrate this expression, we need to consider the variables x and y as constants. So, we can rewrite the integral as:
∫ (2xy^4ey - 2xy^3 + y) dx = 2y^4ey ∫ x dx - 2y^3 ∫ x dx + y ∫ dx
Using the power rule of integration, we can integrate each term separately:
∫ x dx = x^2/2
∫ dx = x
Substituting these results back into the integral, we get:
2y^4ey ∫ x dx - 2y^3 ∫ x dx + y ∫ dx = 2y^4ey * (x^2/2) - 2y^3 * (x^2/2) + y * x
Simplifying further, we have:
= y^4eyx^2 - y^3x^2 + yx
Step 3: Finding the integrating factor
The integrating factor is given by:
IF = e^( ∫ P(x) dx ) = e^(y^4eyx^2 - y^3x^2 + yx)
Therefore, the correct option is c) 1/y^4.
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The integrating factor of the DE(2xy4ey + 2xy3 + y) dx +(x2y4ey - x2y2 - 3x)dy = 0a)1/x4b)y2/x4c)1/y4d)None of theseCorrect answer is option 'C'. Can you explain this answer?
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