A Bernoulli differential equation is one of the form dy/dx+P(x)y=Q(x)yn () Obser

Question

A Bernoulli differential equation is one of the form

dy/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 equation

du/dx+(1?n)P(x)u=(1?n)Q(x)


Consider the initial value problem

xy'+y=?4xy2 y(1)=?5

(a) This differential equation can be written in the form () with
P(x)= .................................,
Q(x)=...................................... , and
n=........................................... .

(b) The substitution u= will transform it into the linear equation

du/dx+................................ u =........................................... .

(c) Using the substitution in part (b), we rewrite the initial condition in terms of x and u:
u(1)=...................................................... .

(d) Now solve the linear equation in part (b). and find the solution that satisfies the initial condition in part (c).
u(x)= ...................................

(e) Finally, solve for y.
y(x)= .................................

Explanation / Answer

(a)P(x)=1/x
Q(x)=4
(b)du/dx+-1/x=-4
u=y^-1
(c)u(1)=-1/5
(d)u/x=-4lnx+c
c=-1/5
u=-x(4ln(x)+1/5))
(e)1/u=-1/[x{4lnx+1/5}]

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