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Chapter 2 Homework Set: Problem 45 (1 point) 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 v = y^{1-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' + 4/x y = 24x^3 y^{1/2} (*) 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(1) = 1 to find C = Finally we obtain the explicit solution of the initial value problem as y =

          Chapter 2 Homework Set: Problem 45
(1 point)
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 v = y^{1-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' + 4/x y = 24x^3 y^{1/2} (*)
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(1) = 1 to find C =
Finally we obtain the explicit solution of the initial value problem as
y =

Chapter 2 Homework Set: Problem 45
(1 point)
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 v = y^1-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' + 4/x y = 24x^3 y^1/2 (*)
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(1) = 1 to find C =
Finally we obtain the explicit solution of the initial value problem as
y =

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