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.