Final answer:
The provided differential equation is shown to be homogeneous, and it is demonstrated that the distance of any point P to the origin is invariant under rotations of the coordinate system.
Step-by-step explanation:
Show that the Equation is Homogeneous
To prove that the differential equation dy/dx = (x² + xy + y²) / x² is homogeneous, we can express each term as a function of one variable by dividing through by x². This yields:
dy/dx = 1 + (y/x) + (y/x)²
Notice that each term is now a function of the variable (y/x), which means that the degree of each term equals the degree of x, verifying the equation as homogeneous.
Distance Invariance Under Rotation
To show that the distance of a point P to the origin is invariant under rotations of the coordinate system, consider a point P with coordinates (x, y) in the original system. After rotation, the coordinates may change to (x', y'), but the squared distance from P to the origin, which is x² + y², remains the same because it only involves the sum of the squares of the coordinates, which is unchanged by rotation. Hence, the distance remains invariant as:
x² + y² = x'² + y'²