A Bernoulli differential equation is one of the form dydx+P(x)y=Q(x)yn () Observ

Question

A Bernoulli differential equation is one of the form

dydx+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=y1n transforms the Bernoulli equation into the linear equation

dudx+(1n)P(x)u=(1n)Q(x).

Consider the initial value problem

xy+y=2xy^2,   y(1)=7.

(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

dy/dx + P(x)*y = Q(x)*y^n

given equation is

xy' + y = 2xy^2

divide by x

y' + (1/x)*y = 2*y^2

divide by y^2

y'/y^2 +(1/x)*(1/y) = 2

A.

P(x) = 1/x

Q(x) = 2

n = 2

B.

Substitute

y^(1 - n) = u

u = 1/y

du/dx = (-1/y^2)*dy/dx

-du/dx + (1/x)*u = 2

du/dx - u/x = -2

C.

y(1) = -7

u(1) = 1/y(1)

u(1) = -1/7

D.

du/dx - u/x = -2

I.F. = exp(integral (-1/x)) = exp(-ln x) = exp(ln (1/x))

I.F. = 1/x

Now solution will be

u*(1/x) = integral (-2*(1/x))*dx

u/x = -2*ln x + C

u/x = ln (1/x^2) + C

u = x*ln (1/x^2) + x*C

Since u(1) = -1/7

-1/7 = 1*ln 1 + C*1

C = -1/7

u = x*ln (1/x^2) - x/7

E.

u = 1/y

y = 1/u

y = 1/(x*ln (1/x^2) - x/7)

y = 7/(7*x*ln (1/x^2) - x)

y = -7/(x*(1 + 14*ln x))

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.