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Find a particular solution Y of the differential equation y³−7y'=4x−3 using the Method of Undetermined Coefficients.

User Yanirys
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Final answer:

To find a particular solution using the Method of Undetermined Coefficients, we assume that the particular solution has the same form as the right-hand side of the equation. Substitute this assumption into the original equation and solve for the coefficients.

Step-by-step explanation:

The given differential equation is y³−7y'=4x−3. To find a particular solution using the Method of Undetermined Coefficients, we assume that the particular solution has the same form as the right-hand side of the equation. Since the right-hand side is a polynomial of degree 1, our assumption becomes Y = Ax + B. Substitute this assumption into the original equation and solve for A and B.

Substituting Y = Ax + B into y³−7y'=4x−3, we get (Ax + B)³−7(A + 0) = 4x−3. Expanding and simplifying, we have A³x³ + 3A²Bx² + 3AB²x + B³ - 7A = 4x−3. Equating like terms on both sides, we get:

  • A³ = 0
  • 3A²B = 0
  • 3AB² = 0
  • B³−7A = −3

Solving these equations, we get A = 1/3 and B = 7/3. Therefore, a particular solution of the given differential equation is Y = (1/3)x + 7/3.

User Osama AbuSitta
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