Mathematics – Final (Global) Exam , BME, January 3, 2017. 90=15+75 minutes. GOOD LUCK!
Theory (3*5=15%.) Please explain your answers and do not forget to formulate the conditions for the functions.
1. Define sinh (sine hyperbolic) function. Sketch its graph, give its Maclaurin series, some identities.
2. What is the derivative of the composed function (chain rule)?
3. What is gradient, Laplacian of a scalar field?
Exercises 25+20+20+20=85%.
1.
Find the solution
y  y  x  the initial value problem
dy y 
2
   x   , y 1  0 . Check it. Sketch the
dx x 
x
f 1 x   yx  , f 2 x  
1
. (10+5+10=25%)
yx 
2. Find the solutions of the initial value problem x(0)  1, y 0   0 for the system of differential equations
x   x  y , y   x  y . Sketch the phase portrait as well. (12+8=20%)




3. Consider the vector field v r   2 x  2 z i  (2 y  z ) j  ( y  x)k .
Find λ if the vector field is potential. Determine the potential function. For this value of λ evaluate the line
  
integral  v r d r (by 2 methods: (i) potential theory, (ii) definition, if  is the straight line connecting the
graph of the functions

points A(0,1,2) and B(-1,2,3)). 4+8+8=20%.
  
For Find λ=0 calculate  v ( r ) dr , if  : x  cos t , y  0, z  sin t ,

(definition, Stokes theorem). (10+10=20%)
0  t  2 by 2 methods