Abernoulli differential equation is one of the formdy/dx + p(x)y = q(x)yn. observe that, if n=0 or 1, the bernoulli equation is linear. for other values of n, the substitution u=y1−n transforms the bernoulli equation into the linear equationdu/dx + (1−n)p(x)u = (1−n)q(x).use an appropriate substitution to solve the equation y′−4xy=y3x2, and find the solution that satisfies y(1)=1.

Asmall (but heavy) particle placed in a glass of water will follow a zigzag motion because the particle will bounce off of the water molecules it meets. this is called brownian motion. a physicist simulates this on a computer, by varying the distance a particle can travel (called the mean free length), on average, before it collides with a water molecule and assigning the change in motion to be one of 8 directions, each with a similar probability. by running the simulated particle (with the same mean free length) many times she determines that it should take 15 seconds, on average, for the particle to fall to the bottom, with a standard deviation of 1.5 seconds. next she lets a real particle fall through a glass of water and finds that it took 18 seconds. what does she conclude, and why?

With number 11 ( show how you got the answer)

As a self-employed seamstress, you know that it takes 7 yards of material to make 3 jackets. you bought 15 yards of material to make 7 jackets. did you buy enough material? if not, how much were you over or under?