Final answer:
To solve the given homogenous equation dy/dx = cos(y/x) + y/x, we can use the substitution method and separate the variables to find the solution.
Step-by-step explanation:
To solve the given homogenous equation dy/dx = cos(y/x) + y/x, we can use the substitution method. Let's substitute y = vx, where v is a function of x. Differentiating both sides of this equation with respect to x, we get dy/dx = v + x(dv/dx). Substituting this back into the original equation, we get v + x(dv/dx) = cos(v) + v. Rearranging the terms and dividing by x, we get (dv/dx) = (cos(v) - v)/x. This is a separable equation, so we can separate the variables and integrate both sides to solve for v. Once we find v, we can substitute it back into the equation y = vx to find the solution for y in terms of x.