8. Consider the dierential equation y\'= y(4 -y). (a) Show that y(t) = 4/(1+Ce^-

Question

8. Consider the dierential equation y'= y(4 -y).

(a) Show that y(t) = 4/(1+Ce^-4t) is a solution for any value of C by plugging it into the

ODE. This family of solutions is called a general solution to the dierential equation.

(b) Sketch the solutions for C = 1,2.....5. (Hint: This ODE is autonomous).

(c) What are the steady-state (constant) solutions?

(d) The general solution may fail to produce all solutions of a dierential equation. Find

a solution that is not given by any value of C. (Hint: Look at part (c)).

(e) Describe a physical situation that this dierential equation could model, and justify

your reasons. (Hint: Consider population growth).

Explanation / Answer

dy/dx = (4-y)y

dy/dx = -(y-4)y

Dividing -

- (dy/dx)/(y-4)y = 1

Integrating both sides w.r.t. x

Integral [(dy/dx)/(y-4)y] dx = integral 1dx

(1/4) log(y) -(1/4) log(4-y) = c + x

or,

y = 4e^(4(c+x))/(e^(4(c+x))+1)

Related Questions

Navigate

• Browse (All)
• Browse 8
• Subjects

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