Differential Equations
2nd Lecture
Instructor: Ahmed Salah Jamal
Tishk International University
Department of Civil Engineering
Fall - 2021
Introduction to Differential
Equations
Course Textbook
2
Zill, D. G. (2018). Advanced engineering mathematics (6th ed.).
Jones & Bartlett Learning, Burlington, Massachusetts, USA.
Grading Criteria
3
• Class Activity and Attendance 10%
• Assignments 10%
• Quizzes 15%
• Midterm Exam. 25%
• Final Exam. 40%
Attention
4
Students should have basic knowledge about:
- Functions and their domains and ranges.
- Differentiation (of nearly all types of functions).
- Different methods of integration.
- Understanding physical problems, the relationships between the variable, and how to translate these relations to
mathematical models.
An equation containing the derivatives of one or more dependent variables, with respect to one or more
independent variables, is said to be a differential equation (DE).
Differential Equations
DE Classification by Type
Ordinary Differential Equations (ODE) Partial Differential Equations (PDE)
An equation containing only ordinary derivatives of
one or more functions with respect to a single
independent variable.
An equation involving only partial derivatives of one
or more functions of two or more independent
variables
Examples:
Examples:
The order of the differential equation (ODE or PDE) is the order of the highest derivative in the equation.The degree of a differential equation is the power (exponent) of the highest order derivative term in the differential equation.
First-order ODE
Second-order ODE
Third-order ODE
Second-order ODE
Fourth-order PDE
DE Classification by Order
Differential Equation Order of DE Degree of DE
32 xdx
dy
36
4
3
3
y
dx
dy
dx
yd
First
Third
First
First
First
03
53
2
2
dx
dy
dx
yd
DE Classification by Linearity
Linear Differential Equations Non-Linear Differential Equations
If the dependent variable y and all its derivatives y’, y’’,
…., y(n) are of the first degree; i.e. the power of each
term involving y is 1 in the DE, then the DE is linear.
Linear differential equation:
1st - order ODE: 𝒂 𝒙 𝐲′ + b 𝒙 𝒚= c 𝒙
2nd - order ODE: 𝒂 𝒙 𝐲′′ + b 𝒙 𝒚′ + c 𝒙 𝐲 = d 𝒙
If the coefficients of y, y’, …., y(n) contain the
dependent variable y or its derivatives or if powers of
y, y’, …., y(n), such as (y’)2, appear in the equation,
then the DE is nonlinear. Also, nonlinear functions of
the dependent variable or its derivatives, such as sin y
or 𝑒𝑦 cannot appear in a linear equation.
Non-Linear differential equation:
1st - order ODE: 𝒚𝐲′ = ‐ 𝒙
2nd - order ODE: (𝐲′′)2 = 𝒚𝒙
Examples:
Examples:
Explicit and Implicit Solutions of Differential Equations
Explicit Solution Implicit Solution
An explicit solution is any solution that is given in the
form y=y(t). In other words, the only place that y
actually shows up is once on the left side and only
raised to the first power.
Let's say that y is the dependent variable and x is the
independent variable. An explicit solution would be
y=f(x), i.e. y is expressed in terms of x only.
An implicit solution is any solution that isn’t in explicit
form.
An implicit solution is when you have f(x,y)=g(x,y)
which means that y and x are mixed together. y is not
expressed in terms of x only. You can have x and y
on both sides of the equal sign or you can have y on
one side and x,y on the other side. An example of
implicit solution is 𝑦 = 𝑥 𝑥 + 𝑦 2
Examples:
Examples:
9
Initial Value Problems
An Initial Value Problem (or IVP) is a differential equation along with an appropriate number of initialconditions that specify the value of the unknown function at a given point in the domain.
forSolve
forSolve
Note that the number of initial conditions required will depend on the order of the differential equation.
Examples:
Solve for
Solve for
General and Particular Solutions of Differential Equations
General Solution
A function which satisfies the given differential equation is called its solution. The general solution to a differential equation is
the most general form that the solution can take and it doesn’t take any initial conditions into account. It should be noted that
in the general solution, the number of arbitrary constants equal to the order of the DE.
1st Order one arbitrary constant
2nd Order two arbitrary constants
3rd Order three arbitrary constants
Examples:
The particular (actual) solution to a differential equation is the specific solution that not only satisfies the differential equation,
but also satisfies the given initial condition(s). That means that it can be obtained from general solution by given particular
values to the arbitrary constants.
Solve for
Solve for
Particular Solution
11