153

CHAPTER 2

Linearity and Nonlinearity

SECTION 2.5

we see that solutions have negative slope

( y′ > 0 )

for 0 < y <

( y′ < 0 )

Nonlinear Models: Logistic Equation

153

when y < 0 or y > a and positive slope b

a . Hence, the equilibrium solution y ( t ) ≡ 0 is unstable, and the equilibrium b

solution y ( t ) ≡ a is stable. b y stable equilibrium

y = a/b

t unstable equilibrium

y= 0

3.

y′ = −ay + by2 , ( a > 0, b > 0 ) We find the equilibrium points by solving y ′ = −ay + by2 = 0 , getting y = 0 ,

a . By inspecting b y′ = y ( −a + by ) ,

a we see that solutions have positive slope when y < 0 or y > a and negative slope for 0 < y < . b b Hence, the equilibrium solution y ( t ) ≡ 0 is stable, and the equilibrium solution y ( t ) ≡ unstable. y

y = a/b y= 0

unstable equilibrium t stable equilibrium

a is b

Solutions manual for differential equations and linear algebra 2nd edition by farlow ibsn 9780134689

Solutions Manual for Differential Equations and Linear Algebra 2nd Edition by Farlow IBSN 9780134689548 Full clear download( no error format...

Solutions manual for differential equations and linear algebra 2nd edition by farlow ibsn 9780134689

Solutions Manual for Differential Equations and Linear Algebra 2nd Edition by Farlow IBSN 9780134689548 Full clear download( no error format...