A Bernoulli differential equation is one of the form Consider the initial value problem (a) This differential equation can be written in the form (*) with P(x) = Q(x) = Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution uy transforms the Bernoulli equation into the linear equation du dx n = (b) The substitution u = du dx + and U= (e) Finally, solve for y. y(x) = dy dx will transform it into the linear equation + P(x)y = Q(x)y" (*) 1-n (c) Using the substitution in part (b), we rewrite the initial condition in terms of x and u: u(1) = + (1-n) P(x)u = (1-n)Q(x). xy + y = 6xy, y(1) = 9. (d) Now solve the linear equation in part (b). and find the solution that satisfies the initial condition in part (c). u(x) =