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(a) Prove that the substitutionJI-n reduces the Bernoulli s equation + P(x)y f(x)y"_ n # 0 . n # 1linear equation in U.Find the solution of the nonlinear IVP d...

Question

(a) Prove that the substitutionJI-n reduces the Bernoulli s equation + P(x)y f(x)y"_ n # 0 . n # 1linear equation in U.Find the solution of the nonlinear IVP dy 2xy dx3y3y(1) = 1

(a) Prove that the substitution JI-n reduces the Bernoulli s equation + P(x)y f(x)y"_ n # 0 . n # 1 linear equation in U. Find the solution of the nonlinear IVP dy 2xy dx 3y3 y(1) = 1



Answers

A Bernoulli differential equation (named after James Bernoulli) is of the form
$ \frac {dy}{dx} + P(x)y = Q(x)y^n $
Observe that, if $ n = 0 $ or 1, the Bernoulli equation is linear. For other values of $ n, $ show that the substitution $ u = y^{1-n} $ transforms the Bernoulli equation into the linear equation
$ \frac {du}{dx} + ( 1 - n)P(x)u = (1 - n)Q(x) $

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