Syllabus

🔹 Unit I – Formation and Basic Types of Differential Equations

Periods: 08 | Marks: 08

1) Formation of Ordinary Differential Equation

Definition:

Differential equation म्हणजे function आणि त्याच्या derivatives मधील संबंध

Example:

$$\frac{dy}{dx} = 2x$$

Formation methods:

• Eliminate arbitrary constants

• Eliminate arbitrary functions

2) Order of Differential Equation

Definition:

Highest order derivative

Example:

$$\frac{d^2y}{dx^2} + 3\frac{dy}{dx} + y = 0$$

Order = 2

3) Degree of Differential Equation

Definition:

Highest power of derivative

Example:

$$\left(\frac{dy}{dx}\right)^2 + y = 0$$

Degree = 2

4) Homogeneous Differential Equation

Definition:

dy/dx = f(y/x)

Example:

$$\frac{dy}{dx} = \frac{x+y}{x}$$

5) Linear Differential Equation

Form:

$$\frac{dy}{dx} + Py = Q$$

Solution method:

Integrating factor

6) Bernoulli’s Equation

Form:

$$\frac{dy}{dx} + Py = Qy^n$$

Convert to linear form

7) Exact Differential Equation

Form:

$$Mdx + Ndy = 0$$

Condition:

$$\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$$

🔹 Unit II – First Order Higher Degree Differential Equations

Periods: 07 | Marks: 07

Types:

1) Solvable for p

Where:

p = dy/dx

2) Solvable for x

Equation expressed in x

3) Solvable for y

Equation expressed in y

4) First Order Higher Degree

Example:

$$\left(\frac{dy}{dx}\right)^2 = x+y$$

🔹 Unit III – Linear Differential Equations with Constant Coefficients

Periods: 08 | Marks: 08

General form:

$$\frac{d^2y}{dx^2} + a\frac{dy}{dx} + by = X$$

Complementary Function (CF)

Solution of homogeneous equation

Particular Integral (PI)

Solution of non-homogeneous equation

General Solution

GS = CF + PI

Homogeneous Linear Differential Equation

Example:

$$\frac{d^2y}{dx^2} + y = 0$$

🔹 Unit IV – Second Order Differential Equations

Periods: 07 | Marks: 07

Wronskian

Used to check linear independence

Formula:

$$W = \begin{vmatrix} y_1 & y_2 \\ y'_1 & y'_2 \end{vmatrix}$$

Change of Dependent Variable

Substitution method

Change of Independent Variable

Variable transformation

Method of Variation of Parameters

Used to find Particular Integral

 To equip students with the knowledge and skills to solve differential equations and apply algebraic techniques to model and analyze the real-world problems.

 To develop analytical and problem-solving skills in the students.