Question

An equation in the form if 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' + 7 / 8y = 5 / 8e -7x y-7 we have n = so v = 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( 0) = 3 1 / 8 to find C = Finally we obtain the explicit solution of the initial value problem as y = y' + p(x)y = q(x)yn n 0,1 v = y1 - n v' + (1 - n)p(x)v = (1 - n)q(x) y' + 7 / 8y = 5 / 8e -7x y-7 y( 0) = 3 1 / 8
An equation in the form with $n\neq 0,1$ is called a Bernoulli equation and it can be solved using the substitution which transforms the Bernoulli equation into the following first order linear equation for :

$\displaystyle{ v'+(1-n)p(x) v = (1-n)q(x)}$

Given the Bernoulli equation

$\displaystyle{ y' + {{\textstyle\frac{7}{8}}}y = {{\textstyle\frac{5}{8}}} e^{ -7 x} y^{-7} }$

we have   so .

We obtain the equation .

Solving the resulting first order linear equation for we obtain the general solution (with arbitrary constant ) given by

Then transforming back into the variables and and using the initial condition to find $C=$ .

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

$\displaystyle{ }$

Answer 1