Department of Mathematics

WORK-BOOKof

Calculus and Differential equations(BSC008)

ForB. Tech. 1st year-2nd semester

Name of the Student:

Enrollment Number:

University of Engineering & Management, Jaipur

Title of Course: Mathematics – Calculus & Differential EquationsCourse Code: BSC008L-T Scheme: 3-1 Course Credits: 4

Introduction:The goal of this mathematics course is to provide high school students and college freshmen an introduction to basic mathematics and especially show how mathematics is applied to solve fundamental engineering problems. The Topics to be covered (tentatively) include:Multivariable Calculus (Integration)Ordinary differential equationsPartial Differential Equations

Course Objectives:The objective of this course is to introduce the basic principles and techniques of Calculus and its engineering applications. It lays the required foundation and skills that can be repeatedly employed in subsequent courses at higher levels. Students will acquire the skills and techniques of:1. Applying double, triple integrations in engineering problems.2. First order and second order ordinary differential equations.3. Computing partial differential equations and their applications to engineering problems.

Learning Outcomes:Knowledge:

1. Student completing the first unit of this course would be expected to find area, volumes of solids and center of mass and gravity.

2. At the end of second unit student will be able to solve first order differential equations.3. After the completion of the third unit, student will be able to solve ordinary differential

equations of second order and problems related to Legendre polynomial.4. At the end of unit 4students will be able to understand, analyze and apply the concept offirst

order linear and non linear partial differential equations.5. After the completion of the fourth unit, student will be able to solve the higher order PDE and

its application in solving wave, diffusion and Laplace equations.6. At the end of this course the student should be able to apply the above mentioned concepts to

engineering problems.

Application:

1. Double and triple integrals and Green’s, Stoke’s theorems can be used to solve physical related applications and one applied in the study of electrical circuits, quantum mechanics and optics.

2. Partial differential equations are used in wave and heat equations.3. Some differential equations cannot be solved using just one function, but can be approximated

as an infinite series (of powers of x).4. At the end of this course the student should be able to apply the above mentioned concepts to

engineering problems.

Course Contents:

Module 1: Multivariable Calculus (Integration): (12 lectures)

Multiple Integration: Double integrals (Cartesian), change of order of integration in double

integrals, Change of variables (Cartesian to polar), Applications: areas and volumes, Triple

integrals (Cartesian); Scalar line integrals, vector line integrals, scalar surface integrals, vector

surface integrals, Theorems of Green, Gauss and Stokes.

Module 2: First order ordinary differential equations: (7 lectures)

Exact, linear and Bernoulli’s equations, Equations of first order and higher degree: equations

solvable for p, equations solvable for y, equations solvable for x and Clairaut’s type.

Module 3: Second order ordinary differential equations and applications: (8 lectures)

Second order linear differential equations with variable coefficients: method of variation of

parameters, Cauchy-Euler equation; General linear ODE of order two with constant coefficients,

C.F. & P.I., D-operator methods for finding P.I.

Module 4: Partial Differential Equations – First order (6 hours)

First order partial differential equations, solutions of first order linear (Lagrange’s Equation) and

non-linear PDEs: four standard forms and Charpit’s method.

Module 5: Partial Differential Equations – Higher order (5 hours)

Separation of variables method to simple problems in Cartesian coordinates. One dimensional

diffusion equation and its solution by separation of variables. Boundary-value problems: Solution

of boundary-value problems for various linear PDEs in various geometries.

Module 1: Multivariable Calculus (Integration):

1. Use divergence theorem to evaluate where F=3 xz i+ y2 j−3 yz k

and the surface of the cube bounded by x=0, y=0, z=0, x=1, y=1, z=1.

2. (a)Show that curl grad f=0 where f=x2y+2xy+z2.

(b) Find ∇ ( ∇ . A )when A= rr .

3. Verify Green’s theorem for where is boundary of the region bounded by x=0, y=0 and x+y=1.

4. A particle moves on the curve x=2t2 ,y=t2-4t, z =3t-5 where t is the time. Find the components of velocity and acceleration at time t=1 in the direction i−3 j+2 k .

5. Verify Stokes theorem for F=(x2+ y2 ) i−2xy j taken round the rectangle bounded by the lines x=±a, y=0, y=b.

6. Evaluate by stoke’s theorem, where and is boundary of the rectangle , & .

7. Evaluate where

8. If where is a constant, show that

9. Evaluate along the curve in the plane from to

.

10. Evaluate the following double integrals:

(i)

(ii) where .

11. Evaluate over the circular disc

12. Evaluate where is a triangle with vertices

13. Evaluate the following integrals:

(i)

(ii)

Module 2: First order ordinary differential equations:

1. Solve

2. Solve

3. Solve .

4. Solve .

5. Solve .

6. Solve .

7. Solve

8. Solve .

9. Prove that ex2

is an I.F. of the equation (x2+x y4 )dx+2 y3dy=0and hence solve it.

10. Solve .

11. solve : ( sinxcosy+e2x )dx+(cosxsiny+ tany)dy=0.

12. Solve: (5 x2+xy−1 )dx+( 12x2− y+2 y2)dy=0 ; giveny=1 when x=0.

Module 3: Second order ordinary differential equations and applications:

1. Solve

2. Solve

3. Solve

4. Solve

5. Solve

6. Solve

7. Solve

8. Solve

9. Solve

10. Solve by using Variation of Parameter method .

11. Solve .

12. Solve by using method of variation of parameters (x2D2−3xD+5 ) y=x2 sinx

13. Solve (x2D2+7 xD+13 ) y=logx

Module 4: Partial Differential Equations – First order:

1. Solve .

2. Solve .

3. Solve .

4. Solve .

5. Solve .

6. Solve .

7. Solve .

8. Solve .

Module 5: Partial Differential Equations – Higher order:

1. Solve ∂u∂ t=5 ∂

2u∂ x2 , u ( x ,0 )=cos5 x ,ux (0 , t )=0 and u(

π2, t )=0 by using separation of variables.

2. Find the bounded solution of ∂u∂ t= ∂

2u∂x2 , where u (0 , t )=1, u ( x ,0 )=0 by using separation of

variables.

3. Solve ∂u∂ t=3 ∂

2u∂ x2 where u(

π2, t )=0,

∂u∂x

¿¿x=0=0 ,u ( x ,0 )=30 cos5 x by using separation of

variables.