3. I n t h i s p r o j e c t , a f i r s t - o r d e r l i n e a r d i f f e r e n t i a l e q u a t i o n
i s s t u d i e d . T h e a p p l i c a t i o n s o f d i f f e r e n t i a l e q u a t i o n s a r e
w r i t e . S o m e e x a m p l e s a n d h o w t o s o l v e t h e m a r e
p r e s e n t e d i n t h i s p r o j e c t .
Abstract
4. Linearity: The dependent variable (y) and its first
derivative (y') appear only to the first power and are
not multiplied together.
Order: The highest derivative involved is the first
derivative.
Integrating Factor: In Mathematics, an integrating
factor is a function used to solve differential
equations. It is a function in which an ordinary
differential equation can be multiplied to make the
function integrable. It is usually applied to solve
ordinary differential equations.
General Solution: The complete set of solutions to
a first-order linear differential equations.
Here is the general form of a first-order linear
differential equation:
πβ²
+ π π π = πΈ π ,
Where π is the unknown function. π¦β²
is the
derivative of π with respect to π. π π and πΈ(π) are
known functions of π.
2. INTRODUCTION
2.1 Overview
A first-order linear differential equation is an
equation that relates an unknown function y of
a variable x, its derivative y', and possibly
other functions of x, in a specific way, and
also can defined as the equations that are
fundamental in many areas of science and
engineering, such as population growth, heat
transfer, and electrical circuits. They can be
solved using a variety of methods, including
separation of variables, integrating factors,
and variation of parameters. First-order linear
differential equations are a fundamental class
of differential equations characterized by:
5. Importance of the linear differential
equation:
1.Building blocks for higher-order linear
differential equations.
2.Wide range of applications in
science, engineering, and finance.
3.Provide a foundation for
understanding more complex differential
equations.
Solving Techniques:
Separation of variables (for
homogeneous).
Integrating factors (for non-
homogeneous).
There are two main types of first-order linear
differential equations:
1- Homogeneous equations: These are equations in
which the right-hand side is zero (Q(x) = 0).
2- Nonhomogeneous equations: These are equations in
which the right-hand side is not zero (Q(x) β 0).
6. Applications of the linear differential equation:
Modeling population growth and decay
Radioactive decay
Chemical kinetics
Electrical circuits
Heat transfer
7. 2.2 Calculation
The general form of the first order of the linear differential equation in the dependent variable π¦ is :
π0(π₯)πβ² + π1(π₯)π¦ = π2(π₯),
where π0(π₯) β 0 , we get
πβ²
+
π1(π₯)
π0(π₯)
π¦ =
π2(π₯)
π0(π₯)
β πβ² + π π₯ π¦ = π π₯ . (1)
where π π₯ =
π1(π₯)
π0(π₯)
πππ π π₯ =
π2(π₯)
π0(π₯)
.
Integrating factor will be a function π(π₯) such that multiplying it to both sides of the equation
π π₯ (π¦β² + π π₯ π¦) = π π₯ π π₯
π π₯ π¦β² + π π₯ π π₯ π¦ = π π₯ π π₯
π π₯ π¦β² + π π₯ π π₯ π¦ = π π₯ π¦β² + πβ² π₯ π¦
π π₯ π¦β²
+ π π₯ π π₯ π¦ ππ₯ = π π₯ π¦ + πΆ.
8. This means the function π should satisfy πβ² π₯ = π π₯ π π₯ , which is separable
differential equation, this can be solved as:
πβ² π₯
π π₯
ππ₯ = π π₯ ππ₯ β
πβ² π₯
π π₯
ππ₯ = π π₯ ππ₯
ln π(π₯) = π π₯ ππ₯ β π π₯ = π π π₯ ππ₯
.
To solve the linear differential equation πβ² + π π₯ π¦ = π(π₯), multiply both sides by the
integrating factor πΌ π₯ = π π π₯ ππ₯
and integrate both sides, or π¦ =
1
πΌ(π₯)
πΌ(π₯)π(π₯) ππ₯ .
9. Example 1: Solve the LDE
ππ¦
ππ₯
=
1
1+π₯3 β
3π₯2
1+π₯2 π¦ ?
Solution:
The above-mentioned equation can
be rewritten as,
ππ¦
ππ₯
+
3π₯2
1+π₯3 π¦ =
1
1+π₯3 .
Comparing it with
ππ¦
ππ₯
+ π(π₯)π¦ =Q(x)
, we get
π(π₯) =
3π₯2
1+π₯3 , and π(π₯) =
1
1+π₯3 .
Letβs figure out the integrating factor
(I.F.) which is π π(π₯)ππ₯. I. πΉ =
π
3π₯2
1+π₯3
ππ₯ = πln 1+π₯3
βπΌ. πΉ. = 1 + π₯
3
10. Now, we can also rewrite the L.H.S as:
π(π¦ Γ πΌ.πΉ)
ππ₯
,
β
π
ππ₯
(π¦ 1 + π₯
3
) =
1
1 + π₯
3 (1 + π₯
3
)
β
π
ππ₯
(π¦ 1 + π₯
3
) = 1
Integrating both the sides w. r. t. x, we get,
β π¦ 1 + π₯
3
= π₯ + π
β π¦ =
π₯
(1 + π₯
3
)
+
π
(1 + π₯
3
)
Example 2: Solve the differential
equation
ππ¦
ππ₯
+ 3π₯2π¦ = 6π₯2 ?
Solution: The given equation is
linear since it has the form of
Equation 1 with
11. πΌ π¦ = π π π¦ ππ¦
= π 2ππ¦
= π2π¦
.
The general solution is
π₯ =
1
πΌ(π¦)
πΌ(π¦)π(π¦) ππ¦
π₯ =
1
π2π¦ 6π2π¦ππ¦ ππ¦
π₯ =
1
π2π¦ 6π3π¦
ππ¦
π₯ =
1
π2π¦ 2π3π¦ + π
π₯ = 2ππ¦ + ππβ2π¦.
Example 3: Solve
ππ¦
ππ₯
=
1
6ππ¦β2π₯
?
Solution: Rewrite the differential
equation.
ππ₯
ππ¦
= 6ππ¦ β2x
ππ₯
ππ¦
+ 2π₯ = ππ¦
It is of the form
ππ₯
ππ¦
+ π(π¦)π₯ = π(π¦).
Here π π¦ = 2 , and π π¦ = ππ¦,
13. Example 5: Solve the differential equation
π
π
π π
π π
β
π
π
ππ =
ππππ π
π
β
π π
π π
β
π
π
π = ππππ
The given equation is linear since it has the form of Equation
1 with π·(π) = β
π
π
and πΈ(π) = ππππ. An integrating factor is
π° π = π β
π
π
π π
= πβππππ = πβπ.
Multiplying both sides of the differential equation
by πβπ, we get
πβπ π π
π π
β
π
ππ π = πππ
or
π
π π
πβπ
π = πππ
Integrating both sides, we have
π₯β4π¦ = πππ₯ππ₯ β¦ β¦ β¦ (1)
πππ‘ π’ = π₯ , ππ£ = ππ₯
ππ₯
β ππ’ = ππ₯ , π£ = ππ₯
π’ππ£ = π’π£ β π£ππ’
πππ₯ππ₯ = π₯ππ₯ β ππ₯ππ₯
= π₯ππ₯ β ππ₯ β¦ β¦ β¦ . (2)
From eq. (1) and eq. (2) , we get
π₯β4π¦ = πππ₯ β ππ₯ + π
π¦ = π₯5ππ₯ β π₯4ππ₯ + π₯4π.
14. S L U
O N I O
S
C C N
Understanding and solving first-order linear differential equations is fundamental
in mathematical modeling. The integrating factor method provides a systematic
approach to find solutions, making these equations valuable tools in analyzing
real-world phenomena. This report provides a brief overview of first-order linear
differential equations, their solution method, and practical applications.
Moreover, some examples are provided to more understand how to solve these
types of differential equations.
15. 4. Reference
[1] Hartman, P. (2002). Ordinary differential equations. Society for Industrial
and Applied Mathematics.
[2] Arnold, V. I. (1992). Ordinary differential equations. Springer Science &
Business Media.
[3] Miller, R. K., & Michel, A. N. (2014). Ordinary differential equations.
Academic press.
[4] Hale, J. K. (2009). Ordinary differential equations. Courier Corporation.