Topics
Mathematical Logic
- Concept of Statements
- Truth Value of Statement
- Logical Connective, Simple and Compound Statements
- Statement Patterns and Logical Equivalence
- Tautology, Contradiction, and Contingency
- Duality
- Quantifier and Quantified 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
- Elementry Transformations
- Properties of Matrix Multiplication
- Application of Matrices
- Applications of Determinants and Matrices
- Overview of Matrices
Trigonometric Functions
- Trigonometric Equations and Their Solutions
- Solutions of Triangle
- Inverse Trigonometric Functions
- Overview of Trigonometric Functions
Pair of Straight Lines
- Combined Equation of a Pair Lines
- Homogeneous Equation of Degree Two
- Angle between lines represented by ax2 + 2hxy + by2 = 0
- General Second Degree Equation in x and y
- Equation of a Line in Space
- Overview of Pair of Straight Lines
Vectors
Line and Plane
- Vector and Cartesian Equations of a Line
- Distance of a Point from a Line
- Distance Between Skew Lines and Parallel Lines
- Equation of a Plane
- Angle Between Planes
- Coplanarity of Two Lines
- Distance of a Point from a Plane
- Overview of Line and Plane
Linear Programming
Differentiation
- Differentiation
- Derivatives of Composite Functions - Chain Rule
- Geometrical Meaning of Derivative
- Derivatives of Inverse Functions
- 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 the Curve, Axis and Line
- Area Between Two Curves
- Overview of Application of Definite Integration
Differential Equations
- Differential Equations
- Order and Degree of a Differential Equation
- Formation of Differential Equations
- Homogeneous Differential Equations
- Linear Differential Equations
- Application of Differential Equations
- Solution of a Differential Equation
- Overview of Differential Equations
Probability Distributions
- Random Variables and Its Probability Distributions
- Types of 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
- Bernoulli Trial
- Binomial Distribution
- Mean of Binomial Distribution (P.M.F.)
- Variance of Binomial Distribution (P.M.F.)
- Bernoulli Trials and Binomial Distribution
- Overview of Binomial Distribution
Notes
A differential equation of the from `(dy)/(dx) + Py = Q`
To solve the first order linear differential equation of the type
`(dy)/(dx) + Py= Q` ...(1)
Multiply both sides of the equation by a function of x say g (x) to get
g(x)`(dy)/(dx) + P.(g(x)) y = Q . g(x)` ...(2)
Choose g(x) in such a way that R.H.S. becomes a derivative of y . g (x).
i.e. `g(x)(dy)/(dx) + P.g(x)y = d/(dx) [y.g(x)]`
or `g(x) (dy)/(dx) + P.g(x)y ` = g(x)`(dy)/(dx) + y g'(x)`
`=> P.g(x) = g'(x)`
o r P = g'(x)/g(x)
Integrating both sides with respect to x, we get
`int Pdx = int (g'(x))/g(x)dx`
or `int P.dx = log (g(x))`
or g(x) = `e^(int Pdx)`
On multiplying the equation (1) by g(x) =`e^( int Pdx)` , the L.H.S. becomes the derivative of some function of x and y. This function
g(x) = `e^(int P dx)` is called Interrating Factor (I.F.) of the given differential equation.
Substituting the value of g (x) in equation (2), we get
`e^(Pdx) (dy)/(dx) + Pe^(int Pdx) y = Q . e^(Pdx)`
Or `d/(dx) (ye^(intPdx)) = Qe^(int Pdx)`
Integrating both sides with respect to x, we get
`y.e^(int P dx) = int (Q.e^(int P dx)) dx` Or
`y = e^(-int Pdx) = int (Q.e^(int P dx)) dx + C`
which is the general solution of the differential equation.
Steps involved to solve first order linear differential equation:
(i) Write the given differential equation in the form `(dy)/(dx)` + Py = Q where P, Q are constants or functions of x only.
(ii) Find the Integrating Factor (I.F) = `e^(int Pdx)`
(iii) Write the solution of the given differential equation as
y (I.F) = `int`(Q × I.F )dx + C
In case, the first order linear differential equation is in the form `(dx)/(dy) + P_1x = Q_1`, where, `P_1` and `Q_1` are constants or functions of y only.
Then I.F = `e^(P_idy)` and the solution of the differential equation is given by
x . (I.F) = `int (Q_1 xx I.F)` dy + C
