Topics and Syllabus - BE Biomedical Engineering Applied Mathematics 2 Semester 2 (FE First Year) University of Mumbai

1.1 Exact differential Equations, Equations reducible to exact form by using integrating factors.
1.2 Linear differential equations (Review), equation reducible to linear form, Bernoulli’s equation.
1.3: Simple application of differential equation of first order and first degree to electrical and Mechanical Engineering problem (no formulation of differential equation)

Concepts :
• Exact Differential Equations  question (5)
• Equations Reducible to Exact Form by Using Integrating Factors  question (3)
• Linear Differential Equations  question (1)
• Equation Reducible to Linear Form 
• Bernoulli’S Equation 
• Simple Application of Differential Equation of First Order and First Degree to Electrical and Mechanical Engineering Problem  question (1)

2.1. Linear Differential Equation with constant coefficient‐ complementary function,
particular integrals of differential equation of the type f(D)y = X where X is 𝑒^𝑎𝑥, sin(ax+b), cos (ax+b), 𝑥^𝑛, 𝑒^𝑎𝑥V, xV.
2.2. Cauchy’s homogeneous linear differential equation and Legendre’s differential equation, Method of variation of parameters

Concepts :
• Linear Differential Equation with Constant Coefficient‐ Complementary Function  question (34)
• Particular Integrals of Differential Equation  question (2)
• Cauchy’S Homogeneous Linear Differential Equation 
• Legendre’S Differential Equation  question (1)
• Method of Variation of Parameters  question (5)

3.1. (a)Taylor’s series method (b)Euler’s method (c) Modified Euler method (d) Runga‐Kutta fourth order formula (SciLab programming is to be taught during lecture hours)
3.2 .Beta and Gamma functions and its properties.

Concepts :
• Taylor’S Series Method  question (4)
• Euler’S Method  question (1)
• Modified Euler Method  question (1)
• Runga‐Kutta Fourth Order Formula  question (3)
• Beta and Gamma Functions and Its Properties  question (1)

4.1. Differentiation under integral sign with constant limits of integration.
4.2. Numerical integration‐ by (a) Trapezoidal (b) Simpson’s 1/3rd (c) Simpson’s 3/8th rule (all with proof). (Scilab programming on (a) (b) (c) (d) is to be taught during lecture hours)
4.3. Rectification of plane curves.

Concepts :
• Differentiation Under Integral Sign with Constant Limits of Integration  question (3)
• Numerical Integration‐ by Trapezoidal  question (2)
• Numerical Integration‐ by Simpson’S 1/3rd  question (2)
• Numerical Integration‐ by Simpson’S 3/8th Rule  question (2)
• Rectification of Plane Curves  question (2)

5.1. Double integration‐definition, Evaluation of Double Integrals.
5.2. Change the order of integration, Evaluation of double integrals by changing the order of integration and changing to polar form.

Concepts :
• Double Integration‐Definition  question (1)
• Evaluation of Double Integrals 
• Change the Order of Integration  question (3)
• Evaluation of Double Integrals by Changing the Order of Integration and Changing to Polar Form 

6.1. Triple integration definition and evaluation (Cartesian, cylindrical and spherical polar coordinates).
6.2. Application of double integrals to compute Area, Mass, Volume. Application of triple integral to compute volume.

Concepts :
• Triple Integration Definition and Evaluation  question (3)
• Application of Double Integrals to Compute Area  question (5)
• Application of Double Integrals to Compute Mass  question (3)
• Application of Double Integrals to Compute Volume  question (1)
• Application of Triple Integral to Compute Volume  question (1)