Topics
Mathematical Logic
- Statements and Truth Values in Mathematical Logic
- Logical Connectives
- Tautology, Contradiction, and Contingency
- Quantifier, Quantified and Duality Statements in Logic
- Negations of Compound Statements
- Converse, Inverse, and Contrapositive
- Algebra of Statements
- Application of Logic to Switching Circuits
- Overview of Mathematical Logic
Matrices
Trigonometric Functions
Pair of Straight Lines
Vectors
Line and Plane
Linear Programming
Differentiation
- Introduction & Derivatives of Some Standard Functions
- Derivative of Composite Functions
- Geometrical Meaning of Derivative
- Derivative of Inverse Function
- Logarithmic Differentiation
- Derivatives of Implicit Functions
- Derivatives of Parametric Functions
- Higher Order Derivatives
- Overview of Differentiation
Applications of Derivatives
- Applications of Derivatives in Geometry
- Derivatives as a Rate Measure
- Approximations
- Rolle's Theorem
- Lagrange's Mean Value Theorem (LMVT)
- Increasing and Decreasing Functions
- Maxima and Minima
- Overview of Applications of Derivatives
Indefinite Integration
Definite Integration
- Definite Integral as Limit of Sum
- Integral Calculus
- Methods of Evaluation and Properties of Definite Integral
- Overview of Definite Integration
Application of Definite Integration
- Application of Definite Integration
- Area Bounded by Two Curves
- Overview of Application of Definite Integration
Differential Equations
Probability Distributions
- Random Variables
- Probability Distribution of Discrete Random Variables
- Probability Distribution of a Continuous Random Variable
- Variance of a Random Variable
- Expected Value and Variance of a Random Variable
- Overview of Probability Distributions
Binomial Distribution
Estimated time: 4 minutes
Maharashtra State Board: Class 12
Definition: Homogeneous Differential Equations
A differential equation of the form \[\frac{dy}{dx}=\frac{f_{1}(x,y)}{f_{2}(x,y)},\] where f1(x, y) ) and f2(x, y) are homogeneous functions of x and y of the same degree, is called a homogeneous differential equation.
Maharashtra State Board: Class 12
Key Points: Homogeneous Differential Equations
(i) Express the homogeneous differential equation in the form
dy/dx = f(x, y) / g(x, y)
(ii) Put y = vx and
dy/dx = v + x dv/dx
Substitute in the equation and cancel out x from the R.H.S.
The equation reduces to the form
v + x dv/dx = F(v)
(iii) Take v on R.H.S. and separate the variables v and x
(iv) Integrate both sides to obtain the solution in terms of v and x
(v) To obtain the required solution in terms of x and y, substitute v = y/x
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