chapter2.page06

# chapter2.page06 - a population can be modelled by I = b t S...

This preview shows page 1. Sign up to view the full content.

6 1. EXPLICITLY SOLVABLE FIRST ORDER DIFFERENTIAL EQUATIONS When g ( y ) is not a constant function, the general solution to y 0 = f ( x ) g ( y ) is given by the equation Z dy g ( y ) = Z f ( x ) dx, (2) which is obtained by dividing both sides of the equation by g ( y ) and then taking antiderivative to both sides. So one has to ﬁnd two antiderivatives R dy g ( y ) and R f ( x ) dx. Some- times we can solution for y ( x ) explicit from the (2), sometimes we can’t ﬁnd explicit formula for y ( x ), in that case we say y ( x ) is an implicitly deﬁned function. Notice, in the process of ﬁnding the general solution to the separable ODE , we divide both sides of the equation by g ( y ), that implicitly requires that g ( y ) 6 = 0 . On the other hand if g ( c ) = 0 then y ( x ) = c is an solution to the separable equation y 0 = f ( x ) g ( y ) This kind solutions are called stationary or steady-state solutions . Sometimes they are also called singular solution . Example 2.2 . In the study of Epidemics, the spread of disease in
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: a population can be modelled by I = ( b ( t ) S ( t )-r ( t )) I where S ( t ) is number of people that is susceptible to the disease but not infected yet. I ( t ) is number of people actually infected. b,r are proportional function. Now suppose S ( t ) = e-3 t , b = t 2 , and r = 4 , ﬁnd all solutions. Solution First notice that I ( t ) ≡ 0 is the steady-state solution. Then, to I = dI dt = ( b ( t ) S ( t )-r ( t )) I, divide its both sides by I , multiplying its both sides by dt and taking antiderivative we have Z dI I = Z b ( t ) S ( t )-r ( t ) dt The left side is easy to ﬁnd, Z dI I = ln | I | + C (3) Plug in the given functions, we get the right hand side Z b ( t ) S ( t )-r ( t ) dt = Z t 2 e-3 t-4 dt...
View Full Document

## This note was uploaded on 10/04/2011 for the course MAP 4231 taught by Professor Dr.han during the Spring '11 term at UNF.

Ask a homework question - tutors are online