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Questions and Answers
Who independently invented calculus in the 17th century?
Who independently invented calculus in the 17th century?
- Gottfried Leibnitz
- Albert Einstein
- Isaac Newton (correct)
- Leonhard Euler
Which mathematician introduced a standard form of linear differential equation of the first order and first degree?
Which mathematician introduced a standard form of linear differential equation of the first order and first degree?
- Euclid
- Isaac Newton
- Pythagoras
- Gottfried Leibnitz (correct)
Which branch of mathematics involves the study of differential equations?
Which branch of mathematics involves the study of differential equations?
- Statistics
- Calculus (correct)
- Geometry
- Algebra
What can be solved using differential equations in game theory?
What can be solved using differential equations in game theory?
Which field does not benefit from the application of differential equations according to the text?
Which field does not benefit from the application of differential equations according to the text?
What distinguishes differential equations from ordinary equations in mathematics?
What distinguishes differential equations from ordinary equations in mathematics?
Which of the following is an example of a practical application of differential equations in real life?
Which of the following is an example of a practical application of differential equations in real life?
What is the purpose of Bernoulli's differential equation?
What is the purpose of Bernoulli's differential equation?
What is the relationship between the independent variable x and the dependent variable y in the given example differential equation $y' = 3x^3$?
What is the relationship between the independent variable x and the dependent variable y in the given example differential equation $y' = 3x^3$?
What is the left-hand side of the given differential equation $y' = 3x^3$ representing?
What is the left-hand side of the given differential equation $y' = 3x^3$ representing?
Which of the following is an example of a practical application of Bernoulli's principle?
Which of the following is an example of a practical application of Bernoulli's principle?
Which of the following is a characteristic of a homogeneous differential equation?
Which of the following is a characteristic of a homogeneous differential equation?
What is the defining characteristic of a non-homogeneous differential equation?
What is the defining characteristic of a non-homogeneous differential equation?
Which type of differential equation is represented by the following function: $y = x^2 + 2xy + y^2$?
Which type of differential equation is represented by the following function: $y = x^2 + 2xy + y^2$?
Which of the following is an example of a non-homogeneous differential equation?
Which of the following is an example of a non-homogeneous differential equation?
What is the primary difference between homogeneous and non-homogeneous differential equations?
What is the primary difference between homogeneous and non-homogeneous differential equations?
Which of the following is an example of a first-order homogeneous differential equation?
Which of the following is an example of a first-order homogeneous differential equation?
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Study Notes
About Mathematicians
- Differential equations have been an important branch of applied mathematics since the 17th century.
- The history of differential equations traces back to calculus, independently invented by Isaac Newton and German scientist Gottfried Leibnitz.
Gottfried Leibnitz
- Introduced a standard form of linear differential equation of the first order and first degree.
- Defined in terms of two variables x and y.
Ordinary Differential Equations (ODE)
- Involve more than one differential term, unlike ordinary equations in mathematics.
- Applications in real life:
- Finding speed and distance or flow of current.
- Motion of an object (solid or liquid), such as a pendulum or ocean waves.
- Explaining thermodynamic concepts.
- Medical applications: checking bacterial decay, growth of population using graphical representation.
Bernoulli's Differential Equation
- A modified version of the Leibnitz equation, overcoming the disadvantage of linear differential equations.
- Practical example: lift generated by an aircraft's wings due to Bernoulli's principle.
Types of Differential Equations
- Ordinary differential equations.
- Partial differential equations.
- Homogeneous differential equations:
- Defined as functions of the same degree.
- Example: homogeneous function in x and y.
- Non-homogeneous differential equations:
- Functions that do not satisfy the property of homogeneity.
- Pendant (non-linear differential equation).
Conclusion
- Calculus, including differential equations, plays a vital role in mathematics and real-life applications.
- Different types of differential equations have various real-life applications.
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