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
- Derivatives of Composite Functions
- Geometrical Meaning of Derivative
- Derivative of Inverse Function
- Logarithmic Differentiation
- Derivative of Implicit Functions
- Derivatives of Functions in Parametric Forms
- 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
- Basic Concepts of Differential Equations
- Order and Degree of a Differential Equation
- Formation of Differential Equations
- Methods of Solving Differential Equations> Homogeneous Differential Equations
- Methods of Solving Differential Equations>Linear Differential Equations
- Applications of Differential Equation
- Solution of a Differential Equation
- Overview of 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
Definition: Linear Differential Equations
A linear differential equation of first order and first degree is
\[\frac{\mathrm{d}y}{\mathrm{d}x}+\mathrm{P}y=\mathrm{Q}\], where P and Q are the functions of x or constants. Its general solution is \[y.\left(\mathrm{I.F.}\right)=\int\mathrm{Q.}\left(\mathrm{I.F.}\right)\mathrm{d}x+\mathrm{c}\] and the function \[\mathrm{e}^{\int\mathrm{Pdx}}\] is called the integrating factor (I.F.) of the given equation.
Method
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Write the given differential equation in standard linear form.
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Identify P and Q.
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Find the integrating factor using \[e^{\int P \, dx}\] or \[e^{\int P \, dy}\].
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Multiply the complete equation by the integrating factor.
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Convert the left-hand side into the derivative of a product.
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Integrate both sides.
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Apply the initial condition, if given, and write the final answer clearly.
Example 1
Solve \[\frac{dy}{dx} - y = \cos x\]
Given equation:
This is already in standard form with P = -1 and Q = cos x.
Integrating factor:
On multiplying throughout by \[e^{-x}\], the equation becomes:
which is equivalent to
Integrating,
or \[\text{I} = - e^{-x} \cos x + \sin x \, e^{-x} - \text{I}\]
or \[2\text{I} = (\sin x - \cos x) \, e^{-x}\]
or \[\text{I} = \frac{(\sin x - \cos x) e^{-x}}{2}$\]
Substituting the value of \[\text{I}\] in equation (1), we get
or \[y = \left( \frac{\sin x - \cos x}{2} \right) + \text{C} e^x\]
which is the general solution of the given differential equation.
Maharashtra State Board: Class 12
Key Points: Linear Differential Equations
- Write the equation in the form dy/dx + Py = Q
- Identify P and Q
- Find I.F. = \[\mathrm{e}^{\int\mathrm{Pdx}}\]
- Multiply the whole equation by I.F.
- Integrate and get a solution.
