Final Answer:
a. The Jacobi determinant from the given differential equation is ∂(y + )/∂y = 1.
b. The equation has multiple solutions. The solution is y = -x^2 + 4x.
Step-by-step explanation:
a. To find the Jacobi determinant, consider the differential equation given: (y + ) Ux + y Uy = 0. For this equation, differentiate with respect to y to find the Jacobi determinant. Taking the partial derivative of y + with respect to y gives ∂(y + )/∂y = 1.
b. Analyzing the differential equation further, it's known that it belongs to a family of curves. By examining the equation, it's revealed that it represents a first-order linear partial differential equation in the form of a characteristic equation. The equation possesses multiple solutions rather than a unique one. It is associated with a family of curves, and considering the initial condition y = 2x, the solution for this specific scenario is found to be y = -x^2 + 4x.
To solve this differential equation, one approach is to rewrite it in terms of total differentials and use separation of variables to find the general solution. Then, applying the initial condition y = 2x allows determining the specific solution for this particular scenario.
Therefore, the Jacobi determinant is ∂(y + )/∂y = 1, signifying that the given equation has multiple solutions. The specific solution derived from the initial condition y = 2x is y = -x^2 + 4x.