Module Week 3
First-Order
Differential Equations
-Homogeneous
Non Separable to
Separable Variables 1
Learning Outcomes
At the end of the lesson, students
able to:
identify homogeneity ODE.
solve by substitution
using separable variables method.
2
NonSe
ODE JAN 2015 Introduction to 2nd Order and Higher
The given family of functions is the general solution of the DE on the
indicated interval. Find a member of the family that is a solution of the IVP.
1.
y c1e x c 2 e x ( , ); y y 0
y (0) 0, y 0) 1
(
2.
y
ModuleWeek3
Linear2ndorderODE
-The Reduction Order-
1
LearningOutcomes
At the end of the lesson, students
able to
use the reduction of order to solve
2nd - order linear ODE.
2
ReductionofOrder
Suppose that y1 is a nontrivial solution
of linear 2nd -order
Ordinary Differential Equations
Test 1 (Solutions)
1. Given the general solution find the particular solution satisfying
[6 marks]
Solution:
Using
Using
The particular solution is
2. Solve the following first order differential equation
Solution: Using Se
ODE JAN 2015 Reduction of Order
The indicated function y1 ( x ) is a solution of the given DE. Use reduction of
order (long method or formula), to find a second solution y 2 ( x ) .
1.
y 4 y 4 y 0;
y1 ( x) e 2 x
2.
y 2 y y 0;
y1 ( x) xe x
3.
y 16 y 0;
4.
ODE Jan 2015 Homogeneous and Bernoulli
Each DE in Problems 1-12 is homogeneous.
Solve the given differential equation by using an appropriate substitution.
1.
(2)
2.
(5)
y
3.
(7)
dy x 3 y
dx 3 x y
4.
(8)
dy y x
dx y x
5.
(9)
ydx ( x xy ) dy 0
6.
(10)
x
2
ModuleWeek3
Linear2ndorderODE
-The Reduction Order-
1
LearningOutcomes
At the end of the lesson, students
able to
reduce 2nd-order linear ODE
to first-order linear ODE.
use the reduction of order to solve
2nd - order linear ODE.
2
2nd-order linear ODE
2
Module3C
Homogeneous 2nd and higher
order Linear Equations with
- Constant Coefficients -
1
LearningOutcome
At the end of the module, students able
to solve the homogeneous linear ODE
with constant coefficients using the
auxiliary equation.
2
Theorem
Supe
Module2Week2
First-Order Ordinary Differential
Equation
- Exact Equation Method -
06/18/15
1
Learning Outcome
At the end of the lesson students
able to
identify a first-order ODE is an
exact equation.
solve using the exact equation
method.
06/18/15
2
De
ModuleWeek5
-The Cauchy-Euler Equation-
LearningOutcome
At the end of the module,
students able to
identify the Cauchy-Euler
equation.
solve the equation.
TheCauchyEulerEquation
A linear DE of the form
n
n 1
d y
y
dy
n 1 d
an x
an 1 x
. a1 x a0 y g ( x)
Module Week 3
2nd and Higher Order Linear
ODE
(Basic Theory)
06/18/15
1
At the of the lesson, students able
to
use theory of linear equations
when solving DE.
06/18/15
2
nth-order linear ODE
Recall
dny
d n 1 y
dy
an ( x) n an 1 ( x) n 1 . a1 ( x) a0 ( x)
ModuleWeek3
First-Order Ordinary Differential
Equation
- Bernoullis Equation-
06/18/15
1
LearningOutcomes
At the end of the lesson,
students able to
solve Bernoullis Equation
using substitution method + Linear
equation method
06/18/15
2
BERNOULLIs EQUATIO
Week1Module2
First-Order Differential Equation
- Linear Equation -
1
Learning Outcomes
At the end of the lesson, students
able to
write linear first-order ODE in
standard form.
solve linear first-order ODE.
2
Definition
A first-order differential equati
Module2Week2
First-Order Differential Equation
- Linear Equation -
1
Learning Outcomes
At the end of the lesson, students
able to
write linear first-order ODE in
standard form.
solve using linear ODE method.
2
Definition
A first-order differential equat
MODULEWEEK1
1.
Overview to ODE
(Terms in ODE)
1
LearningOutcomes
At the end of the lesson, students able to
define what is differential equation.
define what is ordinary differential equation.
identify the order of given ODE.
identify type of variable in
Module 2forWeek1
1st order
ODE
-Method of Separable
Variables 1
Learning Outcomes
At the end of the lesson, students
able to:
solve using separable variables
method (SVM).
2
Reminder!
In solving differential equations,
must able to integrate and some
requ