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(Solved): (15 points) A Bernoulli differential equation is one of the form (dy)/(dx)+P(x)y=Q(x)y^(n) Observe t ...
(15 points) A Bernoulli differential equation is one of the form
(dy)/(dx)+P(x)y=Q(x)y^(n)Observe that, if
n=0or 1 , the Bernoulli equation is linear. For other values of
n, the substitution
u=y^(1-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=-5xy^(2),y(1)=-2.(a) This differential equation can be written in the form
(**)with
P(x)=
Q(x)=
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
xand
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)=
?