An equation in the form y′+p(x)y=q(x)yn with n≠0,1 is called a Bernoulli equation and it can be solved using the substitution v=y1−n which transforms the Bernoulli equation into the following first order linear equation for v: v′+(1−n)p(x)v=(1−n)q(x) Given the Bernoulli equation y′+8xy=28x4y34 (∗) we have n= 3/4 so v= y^(-1/4) . We obtain the equation v′+ v= . Solving the resulting first order linear equation for v we obtain the general solution (with arbitrary constant C) given by v= Then transforming back into the variables x and y and using the initial condition y(1)=1 to find C= . Finally we obtain the explicit solution of the initial value problem as y= Note: You can earn partial credit on this problem.