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Questions and Answers
What is the classification of problems that focus on separating variables in differential equations?
What is the classification of problems that focus on separating variables in differential equations?
Which type of problem typically requires substituting or eliminating constants to simplify the equation?
Which type of problem typically requires substituting or eliminating constants to simplify the equation?
In the context of differential equations, which problem category is primarily concerned with practical implementations or consequences?
In the context of differential equations, which problem category is primarily concerned with practical implementations or consequences?
Which classification could involve solving an equation that has reached a solution without using direct integration?
Which classification could involve solving an equation that has reached a solution without using direct integration?
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What approach is typically used to handle differential equations that can be integrated after rearranging terms?
What approach is typically used to handle differential equations that can be integrated after rearranging terms?
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Study Notes
Differential Equations Past Board Exam
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Problem 1: Order and Degree
- The differential equation is 2x(d4y/dx4) + 5x2(dy/dx)3 - xy = 0.
- The order is fourth.
- The degree is first.
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Problem 2: Eliminating Arbitrary Constants (CE Board November 1994)
- The general solution is y = C₁x + C₂ex.
- The differential equation is (x - 1)y" - xy' + y = 0.
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Problem 3: Eliminating Arbitrary Constants (ECE Board April 1998)
- The general solution is y2 = cx.
- The differential equation is y' = y/2x
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Problem 4: Variable Separable (EE Board October 1997)
- The differential equation is dy = x2dx.
- The curve passes through (1, 1).
- The equation of y in terms of x is x3 - 3y + 2 = 0.
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Problem 5: Variable Separable (EE Board October 1997)
- The differential equation is dy - x dx = 0.
- The curve passes through (1, 0).
- The equation is x2 - 2y - 1 = 0.
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Problem 6: Variable Separable (EE Board October 1995)
- The differential equation is y' = y sec x.
- The general solution is y = C (sec x + tan x).
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Problem 7: Exact DE
- The exact differential equation is 2xy dx + (2 + x2) dy = 0.
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Problem 8: Applications (EE Board April 1997)
- Radium decomposes at a rate proportional to the amount at any instant.
- In 100 years, 100 mg of radium decomposes to 96 mg.
- The amount left after 100 years is 92.16 mg.
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Problem 9: Applications
- Population doubles in 50 years.
- Population increases proportionally to the number of inhabitants
- It will take 116 years to be five times as much.
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Problem 10: Applications (Newton's Law of Cooling)
- The rate of cooling is proportional to the difference in temperatures.
- Air temperature is 30°.
- Substance cools from 100° to 70° in 15 minutes.
- Time to cool from 100° to 50° is 33.59 minutes.
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Description
Test your knowledge of differential equations with this quiz featuring problems from past board exams. It covers topics such as order and degree, eliminating arbitrary constants, and variable separability. Prepare to tackle both theoretical concepts and practical applications.
