17 Questions
Who independently invented calculus in the 17th century?
Isaac Newton
Which mathematician introduced a standard form of linear differential equation of the first order and first degree?
Gottfried Leibnitz
Which branch of mathematics involves the study of differential equations?
Calculus
What can be solved using differential equations in game theory?
Optimizing player strategies
Which field does not benefit from the application of differential equations according to the text?
Social sciences
What distinguishes differential equations from ordinary equations in mathematics?
Differential equations involve more than one differential term.
Which of the following is an example of a practical application of differential equations in real life?
Finding the speed and distance of an object in motion.
What is the purpose of Bernoulli's differential equation?
It is a modified version of the Leibnitz equation that overcomes the disadvantages of linear differential equations.
What is the relationship between the independent variable x and the dependent variable y in the given example differential equation $y' = 3x^3$?
y is a dependent function of x.
What is the left-hand side of the given differential equation $y' = 3x^3$ representing?
The derivative of y with respect to x.
Which of the following is an example of a practical application of Bernoulli's principle?
Lift generated by the wings of an aircraft.
Which of the following is a characteristic of a homogeneous differential equation?
The equation satisfies the property where the function is homogeneous of the same degree in the dependent and independent variables.
What is the defining characteristic of a non-homogeneous differential equation?
The equation does not satisfy the property of homogeneity in the dependent and independent variables.
Which type of differential equation is represented by the following function: $y = x^2 + 2xy + y^2$?
Homogeneous differential equation
Which of the following is an example of a non-homogeneous differential equation?
$y'' + y = \(sin(x)$
What is the primary difference between homogeneous and non-homogeneous differential equations?
Homogeneous equations satisfy the property of homogeneity, while non-homogeneous equations do not.
Which of the following is an example of a first-order homogeneous differential equation?
$y' = \frac{y}{x}$
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.
Test your knowledge on different types of differential equations such as ordinary and partial differential equations. Explore concepts from 'Higher Engineering Mathematics' by B.S. Grewal and 'Ordinary and Partial Differential Equations' by S. Chand.
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