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
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.
Method
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Rewrite the differential equation in a simplified form.
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Check whether the right-hand side depends only on \[\frac{y}{x}\] or \[\frac{x}{y}\].
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Choose substitution: y = vx or x = vy.
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Differentiate the substitution correctly using the product rule.
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Reduce the equation to a separable form.
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Integrate both sides.
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Replace v by \[\frac{y}{x}\] or \[\frac{x}{y}\] to obtain the final answer.
Example 1
Example 11 – Stepwise
Question: Show that \[x \cos\left(\frac{y}{x}\right)\frac{dy}{dx} = y \cos\left(\frac{y}{x}\right) + x\] is homogeneous and solve it.
Step 1: Write in standard form
So it is of the form \[\frac{dy}{dx} = F(x, y)\].
Step 2: Check homogeneity
Let
\[F(x, y) = \frac{y \cos\left(\frac{y}{x}\right) + x}{x \cos\left(\frac{y}{x}\right)}\]
Then
Step 3: Substitution
Put
Substitute into the equation:
So
Hence
Step 4: Integration
Step 5: Back substitution
Replace \[v = \frac{y}{x}\]:
Maharashtra State Board: Class 12
Key Points: Homogeneous Differential Equations
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Check homogeneity first.
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Differentiate substitution carefully.
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Convert to separable form.
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Back-substitute to original variables.
