LINEAR DIFFERENTIAL EQUATION AND BERNOULLIS EQUATIONS
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This document discusses linear differential equations and Bernoulli's equations. It defines an integrating factor as a function that can make a differential equation exact by multiplying both sides. A linear differential equation is one where the derivative term is linear in the dependent variable. The document provides an example of solving a linear differential equation using an integrating factor. Bernoulli's equations are nonlinear differential equations of a specific form, and the document describes how to change variables to transform a Bernoulli's equation into a linear equation that can be solved.
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It provides the method of solving such equations by first reducing it to a linear equation with constant coefficients, then taking a trial solution of the form y = emz, and finally solving the resulting auxiliary equation.
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ENGINEERING THERMODYNAMICS(Basics concept of thermodynamics)
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ENGINEERING THERMODYNAMICS(Basics concept of thermodynamics)
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UNLOCKING HEALTHCARE 4.0: NAVIGATING CRITICAL SUCCESS FACTORS FOR EFFECTIVE I...
The Fourth Industrial Revolution is transforming industries, including healthcare, by integrating digital,
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population health. The study explores stakeholders' perceptions on critical success factors, identifying
challenges such as insufficiently trained personnel, organizational silos, and structural barriers to data
exchange. Facilitators for integration include cost reduction initiatives and interoperability policies.
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Literature Review Basics and Understanding Reference Management.pptx
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CHINA’S GEO-ECONOMIC OUTREACH IN CENTRAL ASIAN COUNTRIES AND FUTURE PROSPECT
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官方认证美国密歇根州立大学毕业证学位证书原版一模一样
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本公司拥有海外各大学样板无数，能完美还原海外各大学 Bachelor Diploma degree, Master Degree Diploma
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留信网认证的作用:
1:该专业认证可证明留学生真实身份
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3:国家专业人才认证中心颁发入库证书
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留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分
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（详情请加一下 文凭顾问+微信:95270640）欢迎咨询！子小伍玩小伍比山娃小一岁虎头虎脑的很霸气父亲让山娃跟小伍去夏令营听课山娃很高兴夏令营就设在附近一所小学山娃发现那所小学比自己的学校更大更美操场上还铺有塑胶跑道呢里面很多小朋友一班一班的快快乐乐原来城里娃都藏这儿来了怪不得平时见不到他们山娃恍然大悟起来吹拉弹唱琴棋书画山娃都不懂却什么都想学山娃怨自己太笨什么都不会斟酌再三山娃终于选定了学美术当听说每月要交元时父亲犹豫了山娃也说爸算了吧咱学校一学期才转
Understanding Inductive Bias in Machine Learning
This presentation explores the concept of inductive bias in machine learning. It explains how algorithms come with built-in assumptions and preferences that guide the learning process. You'll learn about the different types of inductive bias and how they can impact the performance and generalizability of machine learning models.
The presentation also covers the positive and negative aspects of inductive bias, along with strategies for mitigating potential drawbacks. We'll explore examples of how bias manifests in algorithms like neural networks and decision trees.
By understanding inductive bias, you can gain valuable insights into how machine learning models work and make informed decisions when building and deploying them.
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This document provides information about the third edition of the magazine "Sthapatya" published by the Association of Civil Engineers (Practicing) Aurangabad. It includes messages from current and past presidents of ACEP, memories and photos from past ACEP events, information on life time achievement awards given by ACEP, and a technical article on concrete maintenance, repairs and strengthening. The document highlights activities of ACEP and provides a technical educational article for members.
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LINEAR DIFFERENTIAL EQUATION AND BERNOULLIS EQUATIONS
- 1. GANDHINAGAR INSTITUTE OF TECHNOLOGY MECHANICAL DEPARTMENT ADVANCE ENGINEERING MATHEMATICS LINEAR DIFFERENTIAL EQUATION AND BERNOULLIS EQUATIONS
- 2. CONTENTS :- • Introduction • Integrating Factor • Linear Differential Equation • Bernoulli’s Equation
- 3. INTEGRATING FACTOR :- • Sometimes, a differential equation is not exact but can be made by exact by multiplying with a suitable function. This function is known as Integrating Factor(IF).
- 4. LINEAR DIFFERENTIAL EQUATION :- • In each term in a differential equation including the derivative in linear in the term of the dependent variable then the equation is linear. • A Differential Equation in the form of ⅆ𝑦 ⅆ𝑥 + 𝑝𝑦 = 𝑄 …(i) Where P and Q are the function of 𝑥, is called a linear differential equation and linear is 𝑦.
- 5. • Solve the equation (i) to obtain the Integrating Factor (I.F.)…. IF = ⅇ 𝑃 ⅆ𝑥 • Multiplying equation (i) by IF, ⅇ 𝑃 ⅆ𝑥 ⅆ𝑦 ⅆ𝑥 + Pⅇ 𝑃 ⅆ𝑥 𝑦 = Qⅇ 𝑃 ⅆ𝑥 ⅆ ⅆ𝑥 [ⅇ 𝑃 ⅆ𝑥 𝑦] = Qⅇ 𝑃 ⅆ𝑥 Integration W.R.T 𝑥 ⅇ 𝑃 ⅆ𝑥 𝑦 = Qⅇ 𝑃 ⅆ𝑥 dx + c (IF) 𝑦 = (𝐼𝐹)Q + c …(ii) Equation (ii) is known as a differential equation.
- 6. EXAMPLE :- ⅆ𝑦 ⅆ𝑥 + 3 𝑦 𝑥 = 𝑆ⅈ𝑛 𝑥 𝑥3 • Solution:- The equation is linear in 𝑦. P = 3 𝑥 , Q = 𝑆ⅈ𝑛 𝑥 𝑥3 IF = 𝑒 3 𝑥 𝑑𝑥 = 𝑒3log𝑥 = ⅇlog 𝑥3 = 𝑥3 Here, 𝑥3 𝑦 = 𝑥3 sin 𝜒 𝑥3 ⅆ𝑥 + 𝐶 = sin 𝑥 ⅆ𝑥 + 𝐶
- 7. - cos 𝑥 + 𝐶 Y = - Cos 𝜒 𝑥3 + 𝐶
- 8. BERNOULLI’S EQUATION :- • The Equation of the form ⅆ𝑦 ⅆ𝑥 + 𝑝𝑦 = 𝑄𝑦 𝑛 … (i) • P and Q are the function of x or constants is nonlinear equation, known as Bernoulli’s Equation. • Divide the equation (i) by 𝑦 𝑛 , 1 𝑦 𝑛 ⅆ𝑦 ⅆ𝑥 + 𝑝 𝑦 𝑛−1 = 𝑄 …(ii)
- 9. Let , 1 𝑦 𝑛−1 = 𝑣 1−𝑛 𝑦 𝑛 ⅆ𝑦 ⅆ𝑥 = 𝑑𝑣 𝑑𝑥 1/𝑦 𝑛 ⅆ𝑦 ⅆ𝑥 = 1/(1 − 𝑛) 𝑑𝑣 𝑑𝑥 Substitute in equation (ii) 1 1−𝑛 𝑑𝑣 𝑑𝑥 + 𝑝𝑣 = 𝑄 𝑑𝑣 𝑑𝑥 + 1 − 𝑛 𝑝𝑣 = 𝑄
- 10. E𝐗𝐀𝐌𝐏𝐋𝐄 ∶ 𝑑𝑦 𝑑𝑥 + 2𝑦 𝑥 = 𝑦2 𝑥2 Now 1 𝑦2 𝑑𝑦 𝑑𝑥 + 1 𝑦 . 2 𝑥 = 𝑥2 Let 1 𝑦 = 𝑣 − 1 𝑦2 𝑑𝑦 𝑑𝑥 = 𝑑𝑣 𝑑𝑥 … (i) Substitute in equation (i), − 𝑑𝑣 𝑑𝑥 + 2 𝑥 𝑣 = 𝑥2 𝑑𝑣 𝑑𝑥 − 2 𝑥 𝑣 = −𝑥2 …(ii) Here, 𝑝 = 2 𝑥 , 𝑄 = −𝑥2
- 11. 𝐼𝐹 = 𝑒 −2 𝑥 𝑑𝑥 = 𝑒−2log𝑥 = ⅇlog 𝑥−2 = 1 𝑥2 1 𝑥2= 1 𝑥2 − 𝑥2 𝑑𝑥 + 𝑐 = − 𝑑𝑥 + 𝑐 = −𝑥 + 𝑐 𝑣 = −𝑥3 +𝑐𝑥2 1 𝑦 = − 𝑥3 +𝑐𝑥2
- 12. THANK YOU …