MATH24 - Long Quiz #1 - 2nd TERM - 20112012

I.                   React on the following statements. (4 points each)

1.     The primitive solution associated with y = Cx² + C² has a degree of 1.

2.     The equation y – e^x = 0 is a solution of y’ = -y².

3.     The equation y = 5e   ^–x is a solution of y’ + y = 0 on the interval (-0].

4.     The function of g(x,y) = x³ is homogeneous with order .

5.     The differential equation dt – dy = 0 is exact when v denotes an arbitrary constant not parameter.

II.                 Show your solution to the following:

1.     Obtain the differential equation of family of circles with fixed radius r and tangent to x-axis. Draw the diagram that it represents and determine the order, degree, linearity and independent variable(s) of the resulting differential equation. (10 points)

2.     Test the exactness for (2xy – tan y)dx + (x² – xsec² y)dy = 0. If exact, use Partial Derivatives using two ways. (15 points)

3.     Find the general solution of the differential equation (x + b)y’ = ax – ny such that a, b, n are constants with n  0, n  -1, a linear differential equation of order one. (20 points)

4.     Show that equation (xⁿyⁿ⁺¹ + ax)dy + (xⁿ⁺¹yⁿ + ay)dx = 0 if n  0  1, can be solved using integrating factor found by inspection. (20 points)

5.     Using determination of integrating factor, find the solution for (2y² + 3xy – 2y + 6x)dx + x(x + 2y – 1)dy = 0. (15 points)