Math 285           Mid Term 1 Practice               Sp, 2017         Manfroi

 

1.  Replace the second order differential equation with two first order
     differential equations

 

      y’’  =   (dy/dt) sin y  =  0        Answer:     dw/dt = w sin y,    dy/dt = w

 

 

2.  Find the Wronskian of the two functions:

 

              y1(t) = exp(3t2 + 8)  and   y2(t) = exp(3t2 ˗ 4)

 

Answer:      Wr = 0

 

 

3.  The existence and uniqueness theorem for linear differential equations ensures that

      the solution of

                               (t ˗ 2)(t+ 3) y''  + t(t˗2)y' + t2 y = (1/(t+5),  y(˗6) = 2, y'(˗6) = 5

 

    exists for what region?

                                             Answer:  (˗∞, ˗5) U (˗5, ˗3) U (˗3,2) U (2,∞)

 

 

4.  Transform the following differential equation into a separable differential equation.

 

                               dy/dt =  (t + 4y) / (5t ˗ y) 

 

Answer:        dv / (v2 ˗ v +1)  =  dt/t      where  v = y/t

 

 

5.  Transform the differential     y' + (sint) y  =  (cos t) y5  into a linear, first

     order differential equation.

 

Answer:   dv/dt ˗ (4 sin t) v  =  ˗ 4 cos t        where  v = y˗4  

 

 

6.  Find the order of the following differential equation:

 

                (y')5  +  3(y')4 + 2(y')3 ˗ (y')2 + 10 y'  =  0      Answer:  1

 

 

7.  Given a slope field.  Identify the corresponding differential equation.

                                                                                                                           

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