Problem 12. (1 point) A Bernoulli differential equation is one of the form dx dy + p(x)y = e(x)y" (*) Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u = y -" transforms the Bernoulli equation into the linear equation du ax + (1-n)P(x)u = (1-n)@(x). Consider the initial value problem xy' ty = 6xy, y(1) = -1. (a) This differential equation can be written in the form (*) with P (x) =. Q(x) = and n= (b) The substitution u = will transform it into the linear equation du dx u = (c) Using the substitution in part (b), we rewrite the initial condi- tion in terms of x and u: u(1) =_ (d) Now solve the linear equation in part (b), and find the so- lution that satisfies the initial condition in part (c). u( x) = (e) Finally, solve for y. y(x) = Answer(s) submitted: Generated by OWeBWork, http://webwork.maa.org, Mathematical Association of America O