An equation in the form y\' + p(x)y = q(x)y^n with n 0,1 is called a Bernoulli e
ID: 1720432 • Letter: A
Question
An equation in the form y' + p(x)y = q(x)y^n with n 0,1 is called a Bernoulli equation and it can be solved using the substitution nu = y^1-n which transforms the Bernoulli equation into the following first ORDER linear equation for nu: nu' + (1 - n)p(x)nu = (1 - n)q(x) Given the Bernoulli equation y' - 7/4y = -5/4e^-7xy^5 we have n = so nu =. We obtain the equation v' + nu = Solving the resulting first order linear equation for nu we obtain the general solution (with arbitrary constant C) given by nu = Then transforming back into the variables x and y and using the initial condition y(0) = 1 to FIND C = .Finally we obtain the explicit solution of the initial value problem as y =Explanation / Answer
n = 5, v = y^(-4)
v' +7v = 5e^(-7x)
=>
e^(7x) v' + 7e^(7x) v = 5
=>
d(e^(7x)v) = 5dx
=>
e^(7x) v = 5x +c
=>
v = (5x+c)e^(-7x)
=>
y^(-4) = (5x+c)e^(-7x)
y(0) = 1
=>
1 = c
=>
y^(-4) = (5x+1)e^(-7x)
=>
y^4 = e^(7x)/ (5x+1)
=>
y = [e^(7x)/ (5x+1)]^(1/4)