Syllabus
🔹 Unit I – Improper Integral and Gamma Function
Hours: 7 | Marks: 7
Topics:
1) Improper Integral (Definition only)
अयोग्य समाकलन (Improper Integral)
Definition:
जेव्हा definite integral मध्ये limit infinite असते किंवा function infinite होते तेव्हा त्या integral ला improper integral म्हणतात.
Example:
∫1∞x21dx
2) Gamma Function (Γ Function)
Definition:
Γ(n)=∫0∞e−xxn−1dx
Properties of Gamma Function:
Γ(n+1) = nΓ(n)
Γ(1) = 1
Γ(1/2) = √π
Examples based on Gamma Function
🔹 Unit II – Beta Function and Jacobian
Hours: 8 | Marks: 8
1) Beta Function
Definition:
B(m,n)=∫01xm−1(1−x)n−1dx
2) Properties of Beta Function
Important Property:
B(m,n)=Γ(m+n)Γ(m)Γ(n)
3) Relation between Beta and Gamma Function
Important relation:
B(m,n)=Γ(m+n)Γ(m)Γ(n)
4) Examples
5) Jacobian
Definition:
Jacobian is determinant used in change of variables
Formula:
J=∂(u,v)∂(x,y)
🔹 Unit III – Formation of Differential Equations
Hours: 8 | Marks: 8
Topics:
1) Formation of Differential Equation
Arbitrary constant eliminate करून equation तयार करणे
2) Order and Degree
Order:
Highest derivative
Degree:
Power of derivative
3) Homogeneous Differential Equation
Form:
dy/dx = f(y/x)
4) Linear Differential Equation
Form:
dxdy+Py=Q
5) Bernoulli’s Equation
Form:
dxdy+Py=Qyn
6) Exact Differential Equation
Condition:
∂y∂M=∂x∂N
🔹 Unit IV – Linear Differential Equations with Constant Coefficients
Hours: 7 | Marks: 7
General form:
dx2d2y+adxdy+by=X
Topics:
1) Complementary Function (CF)
Solution of homogeneous equation
2) Particular Integral (PI)
Solution of non-homogeneous equation
3) General Solution
General Solution = CF + PI
4) Homogeneous Linear Differential Equation
Example:
dx2d2y+y=0
