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
- Elementry Transformations
- Adjoint of a Matrix
- Application of Matrices
- Overview of Matrices
Trigonometric Functions
- Trigonometric Equations and Their Solutions
- Solutions of Triangle>Polar Co-Ordinates
- Solving a Triangle>Solving a Triangle
- Basics of Inverse Trigonometric Functions
- Graphs of Inverse Trigonometric Functions
- Domain, Range & Principal Value
- Properties of Inverse Trigonometric Functions > Self-adjusting Property
- Overview of Trigonometric Functions
- Properties of Inverse Trigonometric Functions > Reciprocal Property
- Properties of Inverse Trigonometric Functions > Complementary Property
- Properties of Inverse Trigonometric Functions > Addition & Subtraction Formula for Inverse Tangent
- Properties of Inverse Trigonometric Functions > Double-angle Property
- Properties of Inverse Trigonometric Functions > Triple-angle Property
- Properties of Inverse Trigonometric Functions > Addition–Subtraction Formula for Inverse Sine & Cosine
- Properties of Inverse Trigonometric Functions > Negative Argument Property
Pair of Straight Lines
Vectors
- Overview of Vectors
- Basic Concepts of Vector Algebra
- Types of Vectors in Algebra
- Algebra of Vectors > Scalar Multiplication
- Algebra of Vectors > Addition & Subtraction of Two Vectors
- Collinearity and Coplanarity of Vectors
- Vectors in Coordinate Geometry
- Components of Vector in Algebra
- Vector Joining Two Points in Algebra
- Section Formula in Vector Algebra
- Product of Two Vectors > Scalar (Dot) Product
- Product of Two Vectors > Vector (Cross) Product
- Direction Ratios, Direction Cosine & Direction Angles in Vector
- Scalar Triple Product
- Vector Triple Product
Line and Plane
Linear Programming
Differentiation
- Introduction & Derivatives of Some Standard Functions
- Derivatives of Composite Functions
- Geometrical Meaning of Derivative
- Derivative of Inverse Trigonometric Function
- Logarithmic Differentiation
- Differentiation 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
Estimated time: 9 minutes
CBSE: Class 12
Maharashtra State Board: Class 12
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.
CBSE: Class 12
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.
CBSE: Class 12
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
\[x \cos\left(\frac{y}{x}\right)\frac{dy}{dx} = y \cos\left(\frac{y}{x}\right) + x \implies \frac{dy}{dx} = \frac{y \cos\left(\frac{y}{x}\right) + x}{x \cos\left(\frac{y}{x}\right)}\]
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
\[F(\lambda x, \lambda y) = \frac{\lambda y \cos\left(\frac{\lambda y}{\lambda x}\right) + \lambda x}{\lambda x \cos\left(\frac{\lambda y}{\lambda x}\right)} = \frac{\lambda\left(y \cos\left(\frac{y}{x}\right) + x\right)}{\lambda\left(x \cos\left(\frac{y}{x}\right)\right)} = \frac{y \cos\left(\frac{y}{x}\right) + x}{x \cos\left(\frac{y}{x}\right)} = F(x, y)\]
So \[F(x, y)\] is homogeneous of degree 0; the equation is homogeneous.
Step 3: Substitution
Put
\[y = vx \implies \frac{dy}{dx} = v + x\frac{dv}{dx}\]
Substitute into the equation:
\[v + x\frac{dv}{dx} = \frac{v \cos v + 1}{\cos v}\]
So
\[x\frac{dv}{dx} = \frac{v \cos v + 1}{\cos v} - v = \frac{1}{\cos v}\]
Hence
\[\cos v \, dv = \frac{dx}{x}\]
Step 4: Integration
\[\int \cos v \, dv = \int \frac{dx}{x}\]
\[\sin v = \ln|x| + \ln|C|\]
\[\implies \sin v = \ln|Cx|\]
Step 5: Back substitution
Replace \[v = \frac{y}{x}\]:
\[\sin\left(\frac{y}{x}\right) = \ln|Cx|\]
CBSE: Class 12
Maharashtra State Board: Class 12
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.
Video Tutorials
Shaalaa.com | Homogeneous Differential Equation
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