Final answer:
To construct a real function v₁(x, y) such that f = u + iv₁ is analytic everywhere, we find the harmonic conjugate of u by integrating the partial derivatives of u with respect to x and y. The real function v₁ is obtained by equating the expressions for v₁ derived from the Cauchy-Riemann equations. By solving the resulting equations, we find that v₁(x, y) = - y³/2 - yx².
Step-by-step explanation:
To construct a real function v₁(x, y) such that f = u + iv₁ is analytic everywhere, we need to find the harmonic conjugate of u. The real function v₁ will have the property that its partial derivatives satisfy the Cauchy-Riemann equations. In this case, u(x, y) = x³ - 3y²x - x + 10, so we can find v₁ by integrating the partial derivatives of u with respect to x and y.
∂u/∂x = 3x² - 3y² - 1
∂u/∂y = -6y - 3x²
Applying the Cauchy-Riemann equations: ∂v₁/∂x = ∂u/∂y, and ∂v₁/∂y = -∂u/∂x
Integrating ∂u/∂x with respect to x, we get v₁ = x³ - y²x - x + h(y). Integrating ∂v₁/∂y with respect to y, we get v₁ = -3xy - y³/2 - yx² + g(x).
Equating the two expressions for v₁, -3xy - y³/2 - yx² + g(x) = x³ - y²x - x + h(y). From this, we can conclude that g(x) = 0 and h(y) = - y³/2. So, the real function v₁(x, y) = - y³/2 - yx² is such that f = u + iv₁ is analytic everywhere.