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Math - 3 Lecture 4

Lecture 4 covers methods for solving first-order differential equations, specifically Bernoulli's equation, and introduces higher-order differential equations. It explains the transformation of Bernoulli equations into linear equations and discusses homogeneous and nonhomogeneous linear differential equations. The lecture also includes examples, exercises, and references for further reading on the topic.

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13 views18 pages

Math - 3 Lecture 4

Lecture 4 covers methods for solving first-order differential equations, specifically Bernoulli's equation, and introduces higher-order differential equations. It explains the transformation of Bernoulli equations into linear equations and discusses homogeneous and nonhomogeneous linear differential equations. The lecture also includes examples, exercises, and references for further reading on the topic.

Uploaded by

amedalsaby
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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‫ جامعة المنصورة االهلية‬- ‫كلية الهندسة‬

MATHEMATICS METHODS FOR


ENGINEERING
LECTURE (4)
Content Lecture 4
➢Solving First order Differential Equation

❖ Bernoulli ODE

➢ Higher Order Differential Equation


Bernoulli equations

The differential equation of the form


(1)

➢ Where n is any real number, is called Bernoulli’s equation.

➢ Note that for 𝑛 = 0 and 𝑛 = 1, equation is linear.

➢ The substitution 𝑢 = 𝑦1−𝑛 [ for 𝑛 ≠ 0 and 𝑛 ≠ 1 ] reduces any equation from (1)
to a linear equation.
Example (5)

Solution 0

2
3

Example (6)

1
Solution 0 𝑦ƴ 𝑦 −3 + 𝑦 −2 = 3 𝑥 2
𝑥

1 𝑑𝑣
= −2 𝑦 −3 𝑦ƴ
𝑑𝑥

−1 1 −2
𝑣ƴ + 𝑣 = 3 𝑥2 𝑣ƴ + 𝑣 = −6 𝑥 2
2 𝑥 𝑥
2

3 𝑥 −2 𝑣 = න −6 𝑑𝑥 + 𝑐

4
Higher order DE

A linear nth-order differential equation of the form

is said to be homogeneous

with 𝑔(𝑥) not identically zero, is said to be nonhomogeneous.


For example,
is a homogeneous linear second-order DE

is a nonhomogeneous linear third-order DE


Differential Operators

𝑑𝑦
= 𝐷𝑦 . The symbol 𝐷 is called a differential operator
𝑑𝑥
Superposition principle For Homogeneous equation

Note
Linear dependence / independence

Wronskian
For linear independent solutions

General solution for Homogeneous DE


Example (1)

The functions 𝑦1 = 𝑒 3𝑥 and 𝑦2 = 𝑒 −3𝑥 are both solutions of the homogeneous linear

𝑦 ′′ − 9𝑦 = 0

𝑦 = 𝑐1 𝑒 3𝑥 + 𝑐2 𝑒 −3𝑥

is the general solution of the DE


Note: for the second order homogenous DE

In the form 1

Let 𝑦 𝑥 = 𝑦1 is a known solution of (1)

We define 𝑦 𝑥 = 𝑢 𝑦1 is a solution
Example (2)
EXERCISES (2) Solve the given Bernoulli differential equation

1 2

3 4

5 6

7 8

8
EXERCISES (2)
Final Solution for odd numbers examples

3 5

7
Reference and Further Reading

Advanced Engineering Mathematics


Peter V. O'Neil. 7thed.

A First Course in Differential Equations with Modeling Applications 11th Edition


Dennis G. Zill
Thank You

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