(1 point) 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′+76y=12e−7xy−5 we have n= -5 so v= y^6 .

We obtain the equation v′+ 49/6 v= 7/2e^(-7x) .

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(0)=21/6 to find C= .

Finally we obtain the explicit solution of the initial value problem as y=