Mathematics Exam > Mathematics Questions > Y(2xy+1)dx +x(1+2xy-x^3y^3)dy=0?
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Y(2xy+1)dx +x(1+2xy-x^3y^3)dy=0?
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Y(2xy+1)dx +x(1+2xy-x^3y^3)dy=0?
Given Equation:
y(2xy + 1)dx + x(1 - 2xy + x^3y^3)dy = 0
Explanation:
Step 1: Identify the type of equation
- The given equation is a first-order linear ordinary differential equation.
Step 2: Separate the variables
- Write the equation in the form of M(x, y)dx + N(x, y)dy = 0.
- In this case, M(x, y) = y(2xy + 1) and N(x, y) = x(1 - 2xy + x^3y^3).
Step 3: Check if the equation is exact
- An equation is exact if ∂M/∂y = ∂N/∂x, where ∂ denotes partial derivative.
- Calculate the partial derivatives: ∂M/∂y = 2xy + 1 and ∂N/∂x = 1 - 2xy + x^3y^3.
- Since ∂M/∂y is not equal to ∂N/∂x, the equation is not exact.
Step 4: Find the integrating factor
- To make the equation exact, we need to find an integrating factor, denoted by μ(x, y).
- Multiply the entire equation by μ(x, y): μ(x, y)[y(2xy + 1)dx + x(1 - 2xy + x^3y^3)dy] = 0.
- We want the coefficients of dx and dy to be equal, so we choose μ(x, y) such that μ(x, y)∂M/∂y = μ(x, y)∂N/∂x.
- In this case, μ(x, y) = 1/(xy).
Step 5: Multiply the equation by the integrating factor
- Multiply the original equation by μ(x, y) = 1/(xy): (1/(xy))[y(2xy + 1)dx + x(1 - 2xy + x^3y^3)dy] = 0.
- Simplifying, we get: (2 + 1/x)dx + (1 - 2y + x^2y^3)dy = 0.
Step 6: Solve the exact equation
- Now that the equation is exact, we can solve it using integration.
- Let F(x, y) be the potential function, such that ∂F/∂x = 2 + 1/x and ∂F/∂y = 1 - 2y + x^2y^3
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Y(2xy+1)dx +x(1+2xy-x^3y^3)dy=0?
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Y(2xy+1)dx +x(1+2xy-x^3y^3)dy=0? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Y(2xy+1)dx +x(1+2xy-x^3y^3)dy=0? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Y(2xy+1)dx +x(1+2xy-x^3y^3)dy=0?.
Y(2xy+1)dx +x(1+2xy-x^3y^3)dy=0? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Y(2xy+1)dx +x(1+2xy-x^3y^3)dy=0? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Y(2xy+1)dx +x(1+2xy-x^3y^3)dy=0?.
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