Calculus And Differential Equations ( कलन एवं अवकल समीकरण) Paper-II

Units

Topics

I

1.1    Historical background:

1.1.1 Development of Indian Mathematics: Ancient and Early Classical Period (till 500 CE)

1.1.2 A brief biography of Bhaskaracharya (with special reference to Lilavati) and Madhava

1.2    Successive differentiation

1.2.1 Leibnitz theorem

1.2.2 Maclaurin’s series expansion

1.2.3 Taylor’s series expansion

1.3    Partial Differentiation

1.3.1 Partial derivatives of higher order

1.3.2 Euler’s theorem on homogeneous functions

1.4    Asymptotes

1.4.1 Asymptotes of algebraic curves

1.4.2 Condition for Existence of asymptotes

1.4.3 Parallel Asymptotes

1.4.4 Asymptotes of polar curves

II

2.1    Curvature

2.1.1 Formula for radius of Curvature

2.1.2 Curvature of origin

2.1.3 Centre of Curvature

2.2 Concavity and Convexity

2.2.1 Concavity and Convexity of curves

2.2.2 Point of Inflexion

2.2.3 Singular point

2.2.4 Multiple points

2.3 Tracing of curves

2.3.1 Curves represented by Cartesian equation

2.3.2 Curves represented by Polar equation

III

3.1    Integration of transcendental functions

3.2    Introduction to Double and Triple Integral

3.3    Reduction formulae

3.4    Quadrature

3.4.1 For Cartesian coordinates

3.4.2 For Polar coordinates

3.5    Rectification

3.5.1 For Cartesian coordinates

3.5.2 For Polar coordinates

IV

4.1     Linear differential equations

4.1.1 Linear equation

4.1.2 Equations reducible to the linear form

4.1.3 Change of variables

4.2     Exact differential equations

4.3     First order and higher degree differential equations

4.3.1 Equations solvable for x, y and P

4.3.2 Equations homogenous in x and y

4.3.3 Clairaut’s equation

4.3.4 Singular solutions

4.3.5 Geometrical meaning of differential equations

4.3.6 Orthogonal trajectories

V

5.1 Linear differential equation with constant coefficients

5.2 Homogeneous linear ordinary differential equations

5.3 Linear differential equations of second order

5.4 Transformation of equations by changing the dependent/ independent variable

5.5 Method of variation of parameters.

bdkbZ

fo"k;

I

1.1.    ,sfrgkfld i`"BHkwfe

1.1.1. Hkkjrh; xf.kr dk fodkl&  çkphu vkSj çkjfEHkd fpjçfrf"Br dky ¼500 lhbZ rd½

1.1.2. HkkLdjkpk;Z ¼yhykorh ds fo'ks"k lUnHkZ esa½ vkSj ek/ko dh laf{kIr thouhA

1.2.   mÙkjksÙkj vodyu

1.2.1. yScuht çes;

1.2.2. eSDykfju Js.kh }kjk foLrkj

1.2.3. Vsyj Js.kh }kjk foLrkj

1.3.   vkaf'kd vodyu

1.3.1. mPp dksfV ds vkaf'kd vodyt

1.3.2. le?kkr Qyuksa ij vk;yj çes;

1.4.  vuUrLi'khZ

1.4.1. chth; oØksa dh vuUrLif'kZ;ksa

1.4.2. vuUrLi'khZ ds vfLrRo gksus dk çfrcU?k

1.4.3. lekUrj vuUrLif'kZ;k¡

1.4.4. /kzqoh; oØksa dh vuUrLif'kZ;k¡

II

2.1.   oØrk

2.1.1. oØrk f=T;k ds fy, lw=

2.1.2. ewy fcUnq ij oØrk

2.1.3. oØrk dsUæ

2.2.   mÙkyrk ,oa voryrk

2.2.1. oØksa dh mÙkyrk ,oa voryrk

2.2.2. ufr ifjorZu fcUnq

2.2.3. fofp= fcUnq

2.2.4. cgqy fcUnq

2.3.     oØksa dk vuqjs[k.k

2.3.1. dkrhZ; lehdj.kksa }kjk fu:fir oØ

2.3.2. /kzqoh; lehdj.kksa }kjk fu:fir oØ

III

3.1.   vchth; Qyuksa dk lekdyu

3.2.    f}d ,oa f=d lekdy dk ifjp;

3.3.    leku;u lw=

3.4.     {ks=Qyu

3.4.1. dkrhZ; funsZ'kkadksa ds fy,

3.4.2. /kzqoh; funsZ'kkadksa ds fy,

3.5.   pkidyu

3.5.1. dkrhZ; funsZ'kkadksa ds fy,

3.5.2. /kzqoh; funsZ'kkadksa ds fy,

IV

4.1.   jSf[kd vody lehdj.k

4.1.1. jSf[kd lehdj.k

4.1.2. jSf[kd lehdj.k esa lekus; vody lehdj.k

4.1.3. pjksa dk ifjorZu

4.2.   ;FkkrFk vody lehdj.k

4.3.   çFke dksfV ,oa mPp ?kkrh; vody lehdj.k

4.3.1.  x, y vkSj p esa gy gksus ;ksX;

4.3.2. x vkSj y esa le?kkr lehdj.k

4.3.3. Dysjks dk lehdj.k

4.3.4. fofp= gy

4.3.5. vody lehdj.kksa ds T;kferh; vFkZ

4.3.6. ykfEcr laNsfn;k¡

V

5.1.     vpj xq.kkadksa okys jSf[kd vody lehdj.k

5.2.   lk/kkj.k jSf[kd le?kkr vody lehdj.k

5.3.   f}rh; dksfV ds jSf[kd vody lehdj.k

5.4.    ijrU=@LorU= pj ds ifjorZu }kjk lehdj.kksa dk :ikUrj.k

5.5.   çkpy fopj.k fof/k