Final answer:
To solve the Bernoulli equation x(dy/dx) - y = 1/y², use the substitution v = y² which transforms it into a linear differential equation. Once v is found, substitute back to get the solution for y.
Step-by-step explanation:
To solve the given differential equation x(dy/dx) - y = 1/y² which is a Bernoulli equation, we can make a substitution to simplify it. The Bernoulli equation is characterized by its non-linear term in y, but we can transform it into a linear equation by introducing an appropriate substitution. Here, because the equation is of the form y'+P(x)y=Q(x)yⁿ¹, we can use the substitution v = y², which gives dv/dx = 2y(dy/dx). Substituting these into the original equation, and dividing through by x, leads to a linear first-order differential equation in v which can often be solved by integrating factors or separation of variables.
Once we find the solution for v, we can then substitute back v = y² to obtain the solution for y. This method works well because it reduces a non-linear differential equation to a linear form that is more straightforward to solve.