Final answer:
To solve the given differential equation x(dy/dx) = 6y by separation of variables, we separate the variables, integrate both sides, and solve for y. The solutions are y = Kx^6 or y = -Kx^6, depending on the sign of x.
Step-by-step explanation:
To solve the given differential equation x(dy/dx) = 6y by separation of variables, we first separate the variables by rewriting the equation as (1/y)dy = 6dx/x. Then, we integrate both sides of the equation. The integral of (1/y)dy is ln|y|, and the integral of 6dx/x is 6ln|x|. So, we have ln|y| = 6ln|x| + C, where C is the constant of integration. To solve for y, we exponentiate both sides and simplify: |y| = e^(6ln|x|+C) = e^(ln|x|^6+C) = e^(ln|x|^6)e^C = K|x|^6, where K = e^C is another constant. Finally, we consider the cases of positive and negative x to obtain the solutions y = Kx^6 or y = -Kx^6.