MID SEMESTER ONLINE EXAMINATION
1ST SEMESTER, 2020/2021 ACADEMIC YEAR
DATE: OCTOBER 2020
COURSE CODE: MATH 105
COURSE TITLE: ENGINEERING MATHEMATICS II
LECTURER NAME: BERNARD OPOKU
INSTRUCTIONS TO CANDIDATES
FOLLOW THE INSTRUCTIONS GIVEN FOR EACH SECTION
_____________________________________________________________________
PART A [Total: 25 Marks]
INSTRUCTIONS: Part A contains FIVE questions. Answer ALL questions.
1. Define the following terminologies in a Differential Equation below;
I.
When do we say an equation is differential?
II.
Degree of a differential equation.
III.
Order of a differential equation.
IV.
Initial – Value Problem and Boundary – Value Problem of Differential
Equation.
V.
State the two types of differential equation, and clearly defined them with an
example of each.
R (5)
2. Determine the general solution for this problem;
𝐝𝐲
=
𝐝𝐱
𝟐𝐱 𝟐
UN (5)
𝐲
3. Solve the differential equations by separation of variables, and compute their
particular solution with a given IVP
𝟗𝒚𝒚′ + 𝟒𝒙 = 𝟎 ; 𝒚(𝟏) = 𝟒
R (5)
4. Show that for any values of the arbitrary constant 𝑐1 and 𝑐2 the function
𝒚 = 𝒄𝟏 𝒄𝒐𝒔 𝒙 + 𝒄𝟐 𝒔𝒊𝒏 𝒙 is a solution of the differential equation
𝒅𝟐 𝒚
𝒅𝒙𝟐
+𝒚=𝟎
R (5)
5. Determine the differential equation whose general solution is
𝒚 = 𝒄𝟏 𝒆𝒙 + 𝒄𝟐 𝒆−𝒙 + 𝟐𝒙.
UN (5)
PART B
INSTRUCTIONS: Part B contains THREE questions. Answer ANY TWO questions.
QUESTION SIX [Total: 25Marks]
In Problems 6(a) through (c), find values 𝑐1 and 𝑐2 so that the given functions will
satisfy the given conditions. Determine or state whether the given conditions are
initial conditions or boundary conditions:
a.
𝒚 = 𝒄𝟏 𝒆𝒙 + 𝒄𝟐 𝒆−𝒙 + 𝟒𝒄𝒐𝒔𝒙 ; 𝒚(𝟎) = 𝟏 , 𝒚′ (𝟏) = −𝟏
b.
𝒚 = 𝒄𝟏 𝒆𝒙 + 𝒄𝟐 𝒙𝒆𝒙 + 𝒙𝟐 𝒆𝒙 𝒚(𝟏) = 𝟏 , 𝒚′ (𝟏) = −𝟏
𝝅
c. 𝒚 = 𝒄𝟏 𝒔𝒊𝒏𝒙 + 𝒄𝟐 𝒄𝒐𝒔𝒙 ; 𝒚(𝟎) = 𝟏 , 𝒚 ( 𝟐 ) = 𝟏
AP (10)
AN (5)
AP (10)
QUESTION SEVEN [Total: 25Marks]
𝟏
a. Show that 𝒚 = 𝟏+𝒙𝟐 is a solution of 𝒚′′ + 𝟐𝒙𝒚𝟐 = 𝟎
AP (5)
b. Write the differential equation of 𝒆𝒙 𝒚′ + 𝒆𝟐𝒙 𝒚 = 𝒔𝒊𝒏𝒙 general form and
standard form.
AN (10)
c. Determine the differential equation of 𝒙𝒚′ + 𝒚𝟐 = 𝟎 in a differential form.
AP (10)
QUESTION EIGHT [Total: 25Marks]
a. Find the particular solution of
𝒅𝒚
𝒙𝟐
= 𝟏+𝒚𝟐 ; 𝒚(𝟐) = 𝟏
𝒅𝒙
AN (10)
b. Show that the function given; 𝒇(𝒙, 𝒚) =
𝟐𝒙+𝒚
𝒙𝟐 𝒚𝟐
is homogeneous.
AN (10)
c. Verify that the differential equation 𝒚 = 𝟐 𝒔𝒊𝒏 𝟐𝒙 is a solution 𝒚′′ + 𝟒𝒚 = 𝟎
AP (5)
PART C
INSTRUCTIONS: Part C contains TWO questions. Answer ANY ONE question.
QUESTION NINE [Total: 25Marks]
a. Solve the differential equation of (𝟐𝒙 + 𝒚)𝒅𝒙 − 𝒙𝒅𝒚 = 𝟎
E (5)
b. Determine whether the differential equation; 𝒇(𝒙, 𝒚) = 𝒙𝟐 + 𝒔𝒊𝒏𝒙 𝒄𝒐𝒔𝒚 is
homogeneous or non-homogeneous.
CR (10)
𝒅𝒚
𝒙
c. Determine the differential equation 𝐥𝐧 𝒙 𝒅𝒙 = 𝒚
CR (10)
QUESTION TEN [Total: 25Marks]
a. Solve 𝒚′ =
𝟐𝒙𝟐
𝟐𝒚
with an initial-value condition 𝒚(𝟏) = 𝟐, find the general and
particular solutions.
b. Prove that 𝒚 = 𝒆𝒙 − 𝒆−𝒙 is a solution of 𝒚′′ = 𝐲 + 𝟐𝒆−𝒙
c. Show that 𝒚 = 𝟐𝒆𝒙 + 𝒙𝒆−𝒙 is a solution of 𝒚′′ + 𝟐𝒚′ + 𝒚 = 𝟎
CR (10)
E (5)
CR (10)