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Ordinary Differential Equations — Concept Maps

Tamil Nadu Board · Class 12 · Mathematics

3 concept maps of Ordinary Differential Equations for Tamil Nadu Board Class 12 Mathematics, each also written out as a text outline.

43 questions24 flashcards3 formulas & key relations5 concepts

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A labeled diagram explaining the definitions of order and degree of a differential equation with illustrative examples.
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3 Concept Maps

Chapter 10: Ordinary Differential Equations — Complete Overview

Chapter 10: Ordinary Differential Equations — Complete Overview

The map in words

  • Ordinary Differential Equations
    • Classification
      • Order
        • Highest Derivative Present
        • Always Defined
      • Degree
        • Power of Highest Derivative
        • Polynomial Form Required
        • NOT DEFINED for sin cos log e
      • ODE vs PDE
        • ODE Single Variable
        • PDE Multiple Variables
      • Linear vs Nonlinear
        • Linear No Products of y
        • Nonlinear Has y-squared or products
    • Formation of DEs
      • From Geometry
        • Eliminate n Constants
        • Differentiate n Times
        • Lines Circles Parabolas Ellipses
      • From Physics
        • Population dN over dt = rN
        • Decay dA over dt = negative kA
        • Cooling dT over dt = k times T minus Tm
    • Solutions
      • General Solution
        • Contains n Arbitrary Constants
        • Order n DE
      • Particular Solution
        • Fixed Constants via Initial Conditions
    • Solution Methods
      • Variables Separable
        • Separate x and y terms
        • Integrate Both Sides
      • Substitution
        • Put z = ax + by + c
        • Reduces to Separable
      • Homogeneous Form
        • dy over dx = g of y over x
        • Substitute y = vx
        • Separable in v and x
      • Linear DE Integrating Factor
        • Type 1 dy over dx + Py = Q
        • IF = e to the power integral P dx
        • Type 2 dx over dy + Px = Q
        • IF = e to the power integral P dy
    • Applications
      • Population Growth
        • Malthusian Law
        • x = x0 times e to kt
      • Radioactive Decay
        • Half Life = ln2 over k
      • Newton Cooling
        • T = Tm + Ce to kt
      • Mixture Problems
        • dx over dt = IN minus OUT

Ordinary Differential Equations — Complete Mind Map

Ordinary Differential Equations — Complete Mind Map

The map in words

  • Ordinary Differential Equations
    • Classification
      • Order
        • Highest derivative
        • Always positive integer
      • Degree
        • Power of highest derivative
        • In polynomial form
        • Not defined with transcendentals
      • ODE vs PDE
        • ODE - one variable
        • PDE - multiple variables
      • Linear vs Nonlinear
        • Linear - first degree
        • Nonlinear - higher powers or products
    • Formation
      • n constants = nth order ODE
      • Differentiate n times
      • Eliminate constants
      • Examples
        • Lines through origin
        • Family of parabolas
        • Family of circles
    • Solutions
      • General Solution
        • n arbitrary constants
        • Represents family of curves
      • Particular Solution
        • Use initial conditions
        • Find value of C
      • Verification
        • Differentiate and substitute
    • Methods
      • Variables Separable
        • Separate x and y terms
        • Integrate both sides
        • Add single constant C
      • Substitution
        • z = ax + by + c
        • Reduces to separable
      • Homogeneous
        • Verify dy over dx = g of y over x
        • Substitute y = vx
        • Separate in v and x
        • Back-substitute v = y over x
      • Linear ODE
        • Standard form dy over dx + Py = Q
        • IF = e raised to integral P dx
        • Solution formula y times IF
    • Applications
      • Population Growth
        • dN over dt = kN
        • Malthusian law
      • Radioactive Decay
        • dA over dt = minus kA
        • Half life concept
      • Newtons Cooling
        • dT over dt = k times T minus Tm
        • Ambient temperature
      • Mixture Problems
        • dx over dt = IN minus OUT
        • Tank problems

Differential Equation Solution Method Decision Tree

Differential Equation Solution Method Decision Tree

The map in words

  • Start: Given DE
    • Can variables be separated?
      • Yes: Use Variables Separable Method dy/y = f(x)dx
        • Separate & Integrate
          • Solve using Integration
            • Initial Condition Given?
              • Yes: Find Particular Solution Determine C
                • Final Answer
              • No: General Solution with arbitrary constant C
      • No: Is it linear form dy/dx + P(x)y = Q(x)?
        • Yes: Use Integrating Factor Method I.F. = e^{∫P dx}
        • No: Is it homogeneous dy/dx = g(y/x)?
          • Yes: Use Substitution v = y/x Convert to separable
          • No: Contains expression like ax+by+c?
            • Yes: Use Substitution z = ax+by+c Convert to separable
            • No: Try other methods: Exact equations, Special forms
              • Consult advanced techniques

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Frequently Asked Questions

What are the important topics in Ordinary Differential Equations for Tamil Nadu Board Class 12 Mathematics?
Key topics in Ordinary Differential Equations include Order and Degree of Differential Equations, Classification: ODE, PDE, Linear, Nonlinear, Homogeneous, Formation of Differential Equations, General Solution and Particular Solution. Study these first, then practise questions on each for the Tamil Nadu Board Class 12 board exam.
What do the concept maps for Ordinary Differential Equations show?
The 3 maps show how the ideas in Ordinary Differential Equations connect: Chapter 10: Ordinary Differential Equations — Complete Overview; Ordinary Differential Equations — Complete Mind Map. Each map is also written out as an outline on this page.
How should I revise Ordinary Differential Equations for the Tamil Nadu Board Class 12 board exam?
Learn the core ideas first, then work through the 43 practice questions on Ordinary Differential Equations. Revise definitions regularly and use flashcards for quick recall before the exam.

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