Ch 09 - Differential Equations12th Mathematics
Definition of a Differential Equation
What mathematical components must be strictly present for an equation to be classified as a Differential Equation?
What mathematical components must be strictly present for an equation to be classified as a Differential Equation?
A differential equation is an equation that mathematically links an independent variable, a dependent variable, and the derivatives of the dependent variable with respect to the independent variable.
- Example:
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Distinction: Ordinary vs. Partial Differential Equations
How do you precisely distinguish between an Ordinary Differential Equation (ODE) and a Partial Differential Equation (PDE)?
How do you precisely distinguish between an Ordinary Differential Equation (ODE) and a Partial Differential Equation (PDE)?
- Ordinary Differential Equation (ODE): Involves derivatives with respect to strictly one independent variable. (This is the sole scope of the Class 12 syllabus).
- Partial Differential Equation (PDE): Involves partial derivatives with respect to multiple independent variables.
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Order of a Differential Equation
Define the 'order' of a differential equation, and state whether it can ever be mathematically undefined.
Define the 'order' of a differential equation, and state whether it can ever be mathematically undefined.
The order of a differential equation is strictly defined as the order of the highest derivative mathematically present in the equation.
Boundary Conditions / Domain:
- The order is always perfectly defined for every valid differential equation.
- The order is strictly a positive integer ().
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Also in this chapter
49 more cards to study in Prism
- 4Degree of a Differential Equation
- 5Condition for Undefined Degree
- 6SOP: Rationalisation for Degree Determination
- 7Distinction: General vs. Particular Solution
- 8Arbitrary Constants in General Solutions
- 9Arbitrary Constants in Particular Solutions
- 10SOP: Forming a DE from a Family of Curves
- 11Geometry Translation: Tangent Slope
- 12Geometric Elimination Shortcut: Circles
- 13Geometric Translation: Tangent Midpoint on Axes
- 14Identification: Variable Separable Form
- 15SOP: Variable Separable Integration
- 16SOP: Logarithm-Folding in Separable DEs
- 17Integration Trick: 1/(x(x+1)) Partial Fraction
- 18Integration Pattern: Logarithmic Derivative
- 19Substitution: Linear Argument Separable DEs
- 20Algebraic Setup: The Exponent Split Trick
- 21Algebraic Setup: Factoring Before Separating
- 22Definition: Homogeneous Function of Degree Zero
- 23Visual Check: Homogeneous Differential Equation
- 24SOP: Homogeneous DE Substitution ($y/x$ form)
- 25SOP: Homogeneous DE ($x/y$ Mirror Form)
- 26Standard First-Order Linear DE
- 27Normalising the Linear DE (The P-Trap)
- 28Integrating Factor (I.F.)
- 29PIE Strategy: General Solution of a Linear DE
- 30The Negative Sign Trap in Integrating Factors
- 31Shortcut: Perfect Derivatives in Linear DEs
- 32Mirror Form of Linear DE ($dx/dy$)
- 33Solution Template: Mirror Linear DE
- 34The 'SHL' Decision Flow Strategy
- 35Verification Trick: Reverse-Engineering Solutions
- 36Exact Differential: Quotient Form
- 37Exact Differential: Product Form $d(xy)$
- 38Exact Differential: Inverse Tangent Sum
- 39Integration Shortcut: Inverse Hyperbolic Tangent
- 40Initial Value Problem at a Mathematical Singularity
- 41Translation: Rate Proportional to Amount
- 42Exponential Growth and Decay Equation
- 43Continuously Compounded Interest DE
- 44Nested Substitution (Log of Log)
- 45Formula: Newton's Law of Cooling
- 46Substitution: Bernoulli's Equation
- 47Exam Strategy: Extension Methods in Boards
- 482nd-Order Linear ODE: Complex Roots mapping
- 492nd-Order ODE Solution: Repeated Real Roots
- 502nd-Order ODE Solution: Distinct Real Roots
- 51Simple Harmonic Motion (SHM) Differential Equation
- 52Integration Shortcut: Constant Trial Solution
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