Differential Equations
Karnataka Board · Class 12 · Mathematics
Flashcards for Differential Equations — Karnataka Board Class 12 Mathematics. Quick Q&A cards covering key concepts, definitions, and formulas.
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See them allWhat is a differential equation? Give an example.
Answer
A differential equation is an equation involving derivatives of the dependent variable with respect to independent variable(variables). Example: dy/dx + y = 0 or x(dy/dx) + y = 0. It contains variable…
What is the difference between ordinary and partial differential equations?
Answer
Ordinary differential equation: involves derivatives with respect to only ONE independent variable. Example: d²y/dx² + dy/dx = 0. Partial differential equation: involves derivatives with respect to MO…
Define the order of a differential equation with examples.
Answer
Order is the order of the highest order derivative in the differential equation. Examples: (1) dy/dx = e^x has order 1, (2) d²y/dx² + y = 0 has order 2, (3) d³y/dx³ + x²(d²y/dx²)³ = 0 has order 3. Ord…
Define the degree of a differential equation. When is degree not defined?
Answer
Degree is the highest power (positive integral index) of the highest order derivative when the differential equation is a polynomial in derivatives. Example: (dy/dx)² + dy/dx = sin²y has degree 2. Deg…
Find the order and degree of: d³y/dx³ + 2(d²y/dx²)² - dy/dx + y = 0
Answer
Order = 3 (highest derivative is d³y/dx³), Degree = 1 (the power of highest order derivative d³y/dx³ is 1). Note that (d²y/dx²)² has power 2, but d²y/dx² is not the highest order derivative.
What is the difference between general solution and particular solution?
Answer
General solution: Contains arbitrary constants equal to the order of the differential equation. Example: y = a sin(x + b) where a, b are arbitrary constants. Particular solution: Free from arbitrary c…
What is a solution curve or integral curve?
Answer
The curve y = φ(x) is called the solution curve (integral curve) of the given differential equation when the function φ(x) satisfies the differential equation. When φ(x) and its derivatives are substi…
What is a differential equation with variables separable? Give the general form.
Answer
A first order differential equation dy/dx = F(x,y) where F(x,y) can be expressed as F(x,y) = g(x)h(y), i.e., dy/dx = g(x)h(y). Here g(x) depends only on x and h(y) depends only on y. Variables can be …
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Sources & Official References
- Karnataka SSLC — kseeb.kar.nic.in
- Dept of Pre-University Education, Karnataka
- National Education Policy 2020 — education.gov.in
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