Prove that every member of the family functions y=(1+c*e^t)/(1-c*e^t) is a solution of the differential equation y'=(1/2)(y^2-1).

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hala718's profile pic

hala718 | High School Teacher | (Level 1) Educator Emeritus

Posted on

y= 1+ce^t / 1-ce^t

or y = u/v

==> y'= u'v-uv' / v^2

    y'= (ce^t)(1-ce^t)-(-ce^t)(1+ce^t)/ (1-ce^t)^2

      = ce^t -(c^2)e^2t +ce^t -(c^2)e^2t / (1-ce^t)^2

     = 2ce^t -2(c^2)et^2t  / (1-ce^t)^2

 = 2ce^t(1-ce^t)/(1-ce^t)^2= 2ce^t/(1-ce^t)

==> y'= 2ce^t/(1-ce^t).....(1)

Now substitute with y'

y'=1/2(y^2-1)= 1/2[(1+ce^t)^2/(1-ce^t)^2]-1

  = (1/2)(1+2ce^t +(c^2)e^2t)/(1-ce^t)^2  - 1

We will substitute 1 with (1-ce^t)^2/(a-ce^t)^2

==> y'= (1/2) (4ce^t)/(1-ce^t)^2

         = (2ce^t)/(1-ce^t)^2.....(2)

From (1) and (2) we norice that y value is 0. Then y is a solution for the equation.

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

First, we'll differentiate the given expression y=(1+c*e^t)/(1-c*e^t). For this reason, we'll use the quotient rule.



We'll open the brackets:


After reducing similar terms, we'll get:


Now, we'll work the right side for the given result of y', by substituting y with it's expression.

y'=(1/2)(y^2-1)=(1/2)[(1+c*e^t)^2/(1-c*e^t)^2 - 1]



y' = (1/2)[(2)(2c*e^t)/(1-c*e^t)^2]


We notice that we've obtained the same expression for y', so the given function is a solution of the differential equation, for ever value of c.

neela's profile pic

neela | High School Teacher | (Level 3) Valedictorian

Posted on

 y = (1+ce^t)/(1-ce^t)...........(1) ,  is solution of y = (1/2)(y^2-1).


Multiplying both sides of (1) by 1-ce^t,  we get:

y(1-ce^t ) = 1+ce^t. ...........(2). Or

y -1 = c(y+1)e^t . Or

c = (y-1)/{(y+1)e^t}.............(3).

Differentiating both sides of (2), we get:

y(1-ce^t)' +y'*(1-ce^t) = (1+ce^t)'

y*(-ce^t) +y'(1-ce^t) = ce^t. Or

y' (1-ce^t) = ce^t+yce^t =  c(1+y) e^t. Or

 y' = c(1+y)e^t / (1-ce^t). Substititutting  here the value of c from (3),

y' = {(y-1)/[(y+1)e^t]}[(1+y^t)e^t](1  -  (y-1) e^t/[(y+1)e^t])

= (y-1){1-(y-1)/(y+1)]


= (y^2-1)/2 which proves the enunciation.


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