### MATHEMATICS PAPER I SYLLABUS DIFFERENTIAL EQUATIONS

```MATHEMATICS PAPER I
SYLLABUS
DIFFERENTIAL EQUATIONS & ANALYTICAL SOLID GEOMETRY
Differential Equations
Unit - 1
Differential Equations of First Order and First Degree
Linear Differential Equations
Differential Equations Reducible to Linear Form
Exact Differential Equations
Integrating Factors
Change of Variable
Total Differential Equations
Simultaneous Total Differential Equations
Equations of the form dx/P = dy/Q = dz/R
( Ii ) Method of Grouping
( ii ) Method of Multipliers
Differential Equations of the First Order but not of the First Degree
Equations Solvable for p
Equations Solvable for y
Equations Solvable for x
Equations that do not contain x ( or y)
Equations Homogeneous in x and y
Equations of the First Degree in x and y - Clairout's Equation
Unit - 2
Higher Order Linear Differential Equations
Solution of Homogeneous Linear Differential Equations of Order n with
Constant Coefficients
Solution of Non-homogeneous Linear Differential Equations with
Constant Coefficients by means of Polynomial Operators.
( ii ) When Q(x) = bxk and P(D) = D - a0, a0 0.
( ii ) When Q(x) = bxk and P(D) = anDn + an-1 Dn-1 + …. + a1 D.
( iii ) When Q(x) = beax
( iv ) When Q(x) = b sinax or bcosax
( v ) When Q(x) = eax V, where V is a function of x
( vi ) When Q(x) = beax and P(a) = 0
( vii ) When Q(x) = x V, where V is a function of x
Method of Undetermined Coefficients
Method of Variations of Parameters
Linear Differential Equations with Non-constant coefficients
The Cauchy-Euler Equation
System of Linear Diffrential Equations
Solution of a System of Linear Equations with Constant Coefficients
An Equivalent Triangular System
Degenerate Case: p1(D)p4(D) - p2(D) p3(D) = 0
(Scope as in Differential Equations and their Applications by Zafar Ahsan, published by
Prentics-Hall of India Private Limited, New Delhi.)
Solid Geometry
Unit - 3
The Plane
Equation of a plane in terms of its intercepts on the axes
Equation of a plane through the given points
Systems of planes,two sides of a plane,length of the perpendicular
Bisectors of angles between two planes
Combined equation of two planes
Orthogonal projection on a plane
The Right Line
Equation of a line
Angle between a line and a plane
The condition that given line may lie in a given plane
The condition that the two given lines are coplanar
Number of arbitrary constants in the equations of straight line,Sets of
conditions which determine a line
Shortest distance between the two lines,The length and the equations
of the line of shortest distance between two straight lines
Length of the perpendicular from a given point to a given line
Intersection of three planes,Triangular Prism
The Sphere
Definition and equation of the sphere
Equation of the sphere through four given points
Plane sections of a sphere,Intersection of two spheres
Equation of a circle,Sphere through a given circle
Intersection of a sphere and a line,power of a point
Tangent plane, plane of contact,polar plane, pole of a plane
Angle of intersection of two spheres,Conditions for the two spheres to
be orthogonal
Radical plane,Coaxial system of spheres,Simplified form of the
equation of two spheres
Unit - 4
Cones
Definitions of a cone,vertex,guidingcurve,generators,Equation of a cone
with a given vertex and guiding curve
Enveloping cone of a sphere, equations of a cone with vertex at origin
are homogeneous
Condition that the general equation of second degree should represent
a cone
Condition that a cone may have three mutually perpendicular
generators
Intersection a line and a quadric cone
Tangent lines andTangent plane at a point,Condition that a plane may
touch a cone
Reciprocal cones,Intersection of two cones with common vertex
Right circular cone-Equation of Right circular cone with a given
vertex,axis and semivertical angle.
Cylinders &Conicoids
Definition and equation of the cylinder whose generators intersect a
given conic and are parallel to a given line.
Enveloping cylinder of a sphere
Right circular cylinder-Equation of Right circular cylinder with a given
The conicoid-Nature of hyperboloid of one sheet
(Scope as in Analytical Solid Geometry by Shanti Narayan, published by S.Chand& Company
Ltd.
I YEAR PRACTICAL QUESTION BANK
Unit – I:
Differential equations of first order and first degree:
2
dy
= y 3e − x .
dx
dy y
2. Solve:
+ = y 2 sin x.
dx x
x
x/ y 
x/ y 
1 − dy = 0.
3. Solve: 1 + e
dx + e


y

4. Solve: (xy sin xy + cos xy ) ydx + ( xy sin xy − cos xy )xdy = 0.
5. Solve: x 2 + y 2 + 2 x dx + 2 ydy = 0.
1. Solve: xy −
(
)
(
)
dy
= 0.
dx
3
2 2
4
7. Solve: xy + y dx + 2 x y + x + y dy = 0.
8. Solve: x 2 ydx − x3 + y 3 dy = 0.
6. Solve: 3e x tan y + 1 − e x sec 2 y
(
) (
( )
)
Equations of the first order but not of the first degree:
9. Solve: p 3 (x + 2 y ) + 3 p 2 ( x + y ) + ( y + 2 x ) p = 0. Solve
)
(
10. Solve: x 2 p 2 − 2 xyp + 2 y 2 − x 2 = 0.
11. Solve: xp 2 − 2 yp + ax = 0.
( )
12. Solve: y = 2 px + tan −1 xp 2 . .
(
)
13. Solve: x 2 = a 2 1 + p 2 .
14. Solve: y = 2 p + 3 p 2 .
15. Solve: ( x − a ) p 2 + ( x − y ) p − y = 0.
16. Solve: sin px cos y = cos px sin y + p.
Applications of first order differential equations:
17. Find the orthogonal trajectories of x 2 + y 2 = cx.
18. Find the orthogonal trajectories of r = c (1 − sin θ ).
1
19. Find the orthogonal trajectories of y = c e x .
1
20. Find the orthogonal trajectories of x 2 3 + y 2 3 = c .
1
Unit – II:
Higher order differential equations:
(
)
22. Solve: (D 2 + 1)y = e − x + cos x + x3 + e x cos x.
x
3x
23. Solve: (D 2 + 1)(D 2 + 4 )y = cos cos
.
2
2
24. Solve: (D 2 + 1)y = cos x + xe 2 x + e x sin x.
21. Solve: D 2 + a 2 y = tan ax.
25. Solve
d2y
dy
+ 3 + 2 y = sin x by the method of undetermined coefficients
2
dx
dx
d 2 y dy
26. Solve
+ + y = x 2 by the method of undetermined coefficients
2
dx
dx
(
2
)
27. Solve D − 2 D − 8 y = 9 xe
(
)
− x + 10e − x by the method of undetermined coefficients
28. Solve D 2 − 3D y = 2e 2 x sin x by the method of undetermined coefficients
29. Solve: y"+3 y '+2 y = 12e x using the method of variation of parameters.
2 − x using the method of variation of parameters.
30. Solve: y"+2 y '+ y = x e
31. Solve: y "+ y = 4 x sin x using the method of variation of parameters.
x
32. Solve: y"−2 y '+ y = e log x using the method of variation of parameters.
2
2
y '+
y = 0 by reduction of order method, given y1 = x.
x
x2
Solve 2 x 2 + 1 y"−4 xy '+4 y = 0 by reduction of order method, given y = x.
1
2
2
Solve y"− y '+
y = x log x by reduction of order method, given y1 = x.
x
x2
Solve x 2 y"+ xy '− y = x 2e − x by reduction of order method, given y = x.
1
3
2
d y
d y
dy
Solve: x 4
+ 2 x3
− x2
+ xy = 1.
3
2
dx
dx
dx
2
2 d y − x dy + 2 y = x log x.
Solve: x
dx
dx 2
2
dy
2d y
Solve: (2 x + 3)
− 2(2 x + 3) − 12 y = 6 x.
dx
dx 2
2
dy
2d y
Solve: (x + 3)
− 4( x + 3) + 6 y = log( x + 3).
dx
dx 2
33. Solve y"−
34.
35.
36.
37.
38.
39.
40.
(
)
Unit – III:
Planes:
41. A variable plane is at a constant distance 3p from the origin and meets the axes in A, B and C.
Show that the locus of the centroid of the triangle ABC is x
−2 + y −2 + z −2 = p −2 .
42. A variable plane passes through a fixed point (a,b,c) and meets the co-ordinate axes in A,B,C.
Show that the locus of the point common to the planes through A,B,C parallel to the coordinate planes is a x + b y + c z = 1.
43. Show that the equation 12 x 2 − 2 y 2 − 6 z 2 − 2 xy + 7 yz + 6 zx = 0 represents a pair of
planes and also find the angle between them.
Right Lines:
44. Find the coordinates of the foot of the perpendicular drawn from the origin to the plane
2 x + 3 y − 4 z + 1 = 0; also find the coordinates of the point, which is the image of the origin
in the plane.
45. Find the equation of the plane through the point (1,1,1) and perpendicular to the line
x − 2 y + z = 2,4 x + 3 y − z + 1 = 0.
46. A square ABCD of diagonal 2a is folded along the diagonal AC, so the planes DAC, BAC
and at right angles. Show that the shortest distance between DC and AB is then 2a/√3 .
47. Find the magnitude and the equations of the line of shortest distance between the two lines:
x − 3 y + 15 z − 9 x + 1 y − 1 z − 9
=
=
;
=
=
.
−7
−3
2
5
2
1
48. Find the length and the equations of the shortest distance line between
5 x − y − z = 0, x − 2 y + z + 3 = 0; 7 x − 4 y − 2 z = 0, x − y + z − 3 = 0.
49. Find the magnitude and the equations of the line of the shortest distance between the lines
x y +1 z − 2
=
=
;5 x − 2 y − 3 z + 6 = 0 = x − 3 y + 2 z − 3.
4
3
2
50. Obtain the coordinates of the points where the shortest distance line between the lines
x − 23 y − 19 z − 25 x − 12 y − 1 z − 5
=
=
;
=
=
. meet them.
−6
−4
3
−9
4
2
Spheres:
51. A variable plane through a fixed point (a,b,c) cuts the co-ordinate axes in the point A,B,C
show that the locus of the centers of the spheres OABC is a x + b y + c z = 2.
52. Find the equation of the sphere through the four points (0,0,0), (–a,b,c,), (a, –b,c,), (a,b, –c,)
53. Obtain the equation of the sphere which passes through the three points (1,0,0), (0,1,0), (0,0,1)
and has its radius as small as possible.
54. Find center and the radius of the circle x 2 + y 2 + z 2 − 2 y − 4 z = 11, x + 2 y + 2 z = 15.
55. Obtain the equation of the sphere having the circle
x 2 + y 2 + z 2 + 10 y − 4 z − 8 = 0, x + y + z = 3 as the great circle.
56. Show that the plane 2 x − 2 y + z + 12 = 0 touches the sphere
x 2 + y 2 + z 2 − 2 x − 4 y + 2 z = 3 and find the point of contact.
57. Obtain the equations of the sphere which pass through the circle
x 2 + y 2 + z 2 − 2 x + 2 y + 4 z − 3 = 0,2 x + y + z = 4 and touches the plane 3 x + 4 y = 14.
58. Show that the polar line of
is the line
x +1 y − 2
=
= z + 3 w.r.t to the sphere x 2 + y 2 + z 2 = 1
2
3
7x + 3 2 − 7 y
z
.
=
=
11
5
−1
59. Find the equation of the sphere that passes through the circle
x 2 + y 2 + z 2 − 2 x + 3 y − 4 z + 6 = 0,3x − 4 y + 5 z − 15 = 0 and cuts the sphere
x 2 + y 2 + z 2 + 2 x + 4 y − 6 z + 11 = 0.
60. Find the limiting points of the co-axial system of spheres
x 2 + y 2 + z 2 − 20 x + 30 y − 40 z + 29 + λ ( 2 x − 3 y + 4 z ) = 0.
Unit – IV:
CONES AND CYLINDERS:
61. Find the equation of the cone whose vertex is the point (1,1,0) and whose guiding curve is
y = 0, x 2 + z 2 = 4.
62. Find the equation of the cone with vertex at (1,2,3) and guiding curve
x 2 + y 2 + z 2 = 4, x + y + z = 1.
63. Find the enveloping cone of the sphere x 2 + y 2 + z 2 − 2 x + 4 z = 1 with its vertex at (1,1,1).
64. Find the equation of the cone whose vertex is at the origin and the direction cosines of whose
generators satisfy the relation 3l 2 − 4m 2 + 5n 2 = 0.
65. Find the equation of the cone which passes through the three co-ordinates axes as well as the
x
y
z x
y z
=
= ; =
= .
1 − 2 3 3 −1 1
66. Show that the equation x 2 − 2 y 2 + 3 z 2 − 4 xy + 5 yz − 6 zx + 8 x − 19 y − 2 z − 20 = 0
two lines
represents a cone with vertex (1,–2,3).
67. Find the angle between then lines of intersection of
x + y + z = 0 and x 2 + yz + xy − 3 z 2 = 0.
68. If the plane 2 x − y + cz = 0 cuts the cone yz + xy + zx = 0 in perpendicular lines, find the
value of c.
69. Find the equations of the lines in which the plane
2 x + y − z = 0 cuts the cone 4 x 2 − y 2 + 3 z 2 = 0.
70. Show that the locus of mid-points of chords of the cone
ax 2 + by 2 + cz 2 + 2 fyz + 2 gxz + 2hxy = 0 drawn parallel to the line x l = y m = z n
is the plane x (al + hm + gn) + y (hl + bm + fn ) + z ( gl + fm + cn) = 0.
71. Find the plane, which touches the cone x 2 + 2 y 2 − 3 z 2 + 2 yz − 5 zx + 3 xy = 0 along the
generator whose direction ratios are (1,1,1).
72. Prove that the cones fyz + gzx + hxy = 0;
fx ± gy ± hz = 0 are reciprocal.
73. Find the equation of the right circular cone whose vertex is (1,–2, –1) , axis the line
x −1 y + 2 z +1
and semi-vertical angle 60°.
=
=
3
4
5
74. Find the equation to the cylinder whose generators are parallel to
x y z
= = and guiding
1 2 3
curve is x 2 + y 2 = 16, z = 0.
75. Find the equation of the enveloping cylinder of the conicoid
generators are parallel to the line
x2 y2 z2
+
+
= 1 whose
a 2 b2 c2
x y z
= = .
l m n
76. Obtain the equation of a cylinder whose generators touch the sphere
x 2 + y 2 + z 2 + 2ux + 2vy + 2 wz + d = 0 whosegenerators are parallel to the line
x y z
= = .
l m n
77. Obtain the equation of the right circular cylinder whose guiding curve is the circle through the
points (1,0,0), (0,1,0), (0,0,1).
78. Find the equation of the right circular cylinder of radius 2 whose axis is
x −1 y − 2 z − 2
=
=
.
2
2
2
79. Find the equation of the right circular cylinder whose axis is
x −1 y −1 z
=
= and passes
2
1
3
through (0,0,3).
80. Prove that the right circular cylinder whose one section is the circle
x 2 + y 2 + z 2 − x − y − z = 0, x + y + z = 1, is
x 2 + y 2 + z 2 − yz − zx − xy = 1.
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