Differential Equations and Modeling — Flashcards
CBSE · Class 12 · Applied Mathematics
24 flashcards for Differential Equations and Modeling (CBSE Class 12 Applied Mathematics) to test yourself on key terms and facts.
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What is a differential equation? Give an example.
Answer
A differential equation is an equation involving derivative(s) of the dependent variable with respect to the independent variable(s). Example: dy/dx + y = x² (involves the derivative dy/dx of y with r…
Define the order of a differential equation and find the order of: d³y/dx³ + x²(d²y/dx²)³ = 0
Answer
The order of a differential equation is the highest ordered derivative involved. For d³y/dx³ + x²(d²y/dx²)³ = 0, the highest derivative is d³y/dx³ (third derivative), so the order is 3.
What is the degree of a differential equation? Find the degree of: (d²y/dx²)² - 3(dy/dx)⁴ = y²
Answer
The degree is the highest power of the highest order derivative when the equation is polynomial in derivatives. For (d²y/dx²)² - 3(dy/dx)⁴ = y², the highest order derivative is d²y/dx² raised to power…
When is the degree of a differential equation not defined? Give an example.
Answer
The degree is not defined when the differential equation is not polynomial in its derivatives. Example: d²y/dx² + y² + e^(dy/dx) = 0 (contains e^(dy/dx), which is not polynomial in dy/dx).
What is the difference between general solution and particular solution of a differential equation?
Answer
General solution: Contains arbitrary constants and represents a family of curves. Example: y = ce^(-x) + 1. Particular solution: No arbitrary constants, obtained by giving specific values to constants…
Verify that y = ae^(2x) is a solution of dy/dx - 2y = 0
Answer
Given: y = ae^(2x) Differentiate: dy/dx = 2ae^(2x) Substitute in equation: dy/dx - 2y = 2ae^(2x) - 2(ae^(2x)) = 2ae^(2x) - 2ae^(2x) = 0 ✓ Therefore, y = ae^(2x) is indeed a solution.
How do you form a differential equation from a given family of curves with n parameters?
Answer
Steps: 1) Start with equation f(x,y,a₁,a₂,...,aₙ) = 0, 2) Differentiate n times to get n additional equations, 3) Eliminate all n parameters from the (n+1) equations to get the differential equation.
Form the differential equation for the family of circles with center at origin: x² + y² = a²
Answer
Given: x² + y² = a² ... (1) Differentiate: 2x + 2y(dy/dx) = 0 Simplify: x + y(dy/dx) = 0 This gives: x + y(dy/dx) = 0 or dy/dx = -x/y This is the required differential equation.
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- CBSE Official — cbse.gov.in
- National Education Policy 2020 — education.gov.in
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