MTH202: ORDINARY DIFFERENTIAL EQUATION (ODE) - FUO - 2019 SESSION - FUOEXAMS.BLOGSPOT.COM

MTH202: ORDINARY DIFFERENTIAL EQUATION (ODE) 2018/2019 SESSION

FEDERAL UNIVERSITY OTUOKE (FUO)

INSTRUCTION: Answer All questions in section A and one question each from section B, C, and D.

SECTION A

1.     QUESTION ONE

a.     What is an ordinary Differential Equation? Differentiate between Ordinary differential equation and partial differential equation; give at least one example each.

b.     Explain the following as it relates to studying Ordinary differential equations

                                                              i.      Order of Differential equations

                                                            ii.      Degree of Ordinary Differential equations

                                                          iii.      Solution of Ordinary Differential equation

c.      To form a differential equation the constants in the function tells …

d.     Form an Ordinary Differential Equation given the function y=c₁x + c₂x² + c₃x³

e.     If y = c₁x + c₂x² + c₃x³ is a solution to a higher order ODE, solve the particular solution using y(-1) = 1, y(-2) = 3 and y(1) = 5

SECTION B

2.     QUESTION TWO

a.     Solve the Bernoulli equation  10x²y⁵

3.     QUESTION THREE

a.     Find the solution for the second-order Cauchy-Euler ODE,  – 2x+ 2y = x⁴ , y(1) = 1/6, Ý(1) = -1/3. Assume x = e^x

SECTION C

4.     QUESTION FOUR

a.     Using the method of variation of parameter, solve  + y = cosecxtanx

b.     If y₁ = 2x, y₂ = cosx and y₃ = 2sin3x are solutions of the higher order ordinary differential equation. Determine the Wronskian and comment on your result.

5.     QUESTION FIVE

a.     Reduce the Ordinary differential equation (x² – x + 1) – (x² + x) + (x +1)y = 0, given y = x

b.     Given that y₁ = x is a solution of (x² + 1) – 2x + 2y = 0. Find a general solution by reducing to first order.

SECTION D

6.     QUESTION SIX

a.     Find the general solution of y^iv + 2yⁱⁱⁱ + 6yⁱⁱ +2yⁱ 5y = 0. If  y = sin x is one of the solutions of the differential equation.

b.     Find the general solution of the differential equation d²y/dx² + 6dy/dx + 5y = 2e^x + 10e^5x

7.     QUESTION SEVEN

a.     Find the general solution of the homogeneous D.E (t² + 3y)dt = 2tydy

b.     Show that (2ty – 3)dt +(t² +4y)dy = 0 Is exact. Hence , obtain the solution of the initial value problem (2ty -3)dt +(t² + 4y)dy = 0, y(1) = 2.

c.      Solve the separable equation (1 – sin 2t)e^2y dy + cos 2t(1+e^2y)dt = 0 – (x² + x) + (x +1)y = 0, given y = x

b.     Given that y₁ = x is a solution of (x² + 1) – 2x + 2y = 0. Find a general solution by reducing to first order.

SECTION D

6.     QUESTION SIX

a.     Find the general solution of y^iv + 2yⁱⁱⁱ + 6yⁱⁱ +2yⁱ 5y = 0. If  y = sin x is one of the solutions of the differential equation.

b.     Find the general solution of the differential equation d²y/dx² + 6dy/dx + 5y = 2e^x + 10e^5x

7.     QUESTION SEVEN

a.     Find the general solution of the homogeneous D.E (t² + 3y)dt = 2tydy

b.     Show that (2ty – 3)dt +(t² +4y)dy = 0 Is exact. Hence , obtain the solution of the initial value problem (2ty -3)dt +(t² + 4y)dy = 0, y(1) = 2.

c.      Solve the separable equation (1 – sin 2t)e^2y dy + cos 2t(1+e^2y)dt = 0