DEPARTMENT OF APPLIED SCIENCE AND HUMANITIES

Assam University, Silchar

Syllabus & Problem Set (2023) ASH 301A (ECE)

Mathematics III

TOPICS: Ordinary Differential Equations: Solution of 1st order and 1st degree differential equations
by exact method. Integrating factors, Leibniz’s linear equation, Bernoulli’s equation. Differential equation
of 1st order and of higher degree, Clairaut’s equation. Differential equations of 2nd and higher order with
constant co-efficients, method of variation of parameters for solving 2nd order differential equations.
Algebraic Structures: Binary Operation, Group, Ring, Integral Domain, Field and their properties.

1. Find the order and degree of the following differential equations. Also classify them as linear and non-
linear: √
dy dy dy dy 2 1/2 d2 y 3 dy 5 2
(i) y = x dx + k dx (ii) y = x dx + a{1 + ( dx ) } (iii) dy = (y + sin x)dx (iv) ( dx 2 ) + x( dx ) + y = x
2 2
dy 2 4/3 d y d y 1/3 dy 1/2 d2 y
(v) {y + x( dx ) } = x dx 2 (vi) ( dx2 ) = (y + dx ) (vii) dx2 + x2 cos y = 0.
2. Find the differential equation of (i) all straight lines passing through the origin (ii) all straight line in
the xy-plane (iii) family of concentric circles centered at the origin (iv) all circles which pass through
the origin and whose centres are on the x axis.
2
d y dy 2 dy
3. Show that Ax2 + By 2 = 1 is the solution of x[y dx 2 + ( dx ) ] = y dx .

Exact ODE
4. Solve (i) (x2 − 4xy − 2y 2 )dx + (y 2 − 4xy − 2x2 )dy = 0 (ii) (x + y)2 dx − (y 2 − 2xy − x2 )dy = 0 (iii)
dy 2x−y+1 xdy−ydx 2 2
dx = x+2y−3 (iv) xdx + ydy + x2 +y 2 = 0 (v) y sin 2xdx − (1 + y + cos x)dy = 0.

dy
5. Find the value of the constant λ such that (2xey + 3y 2 ) dx + (3x2 + λey ) = 0 is exact. Further, solve
the equation for this value of λ.
Integrating Factor

6. Solve (i) ydx − xdy + (1 + x2 )dx + x2 sin ydy = 0 (ii) y(2xy + ex )dx = ex dy (iii) 2
p y sin 2xdx = (1 + y +
2 3 2 2 3 2 2 2 dy 2 2
cos x)dy (iv) (x + xy + a y)dx + (y + yx − a x)dy = 0 (v) x dx + xy = 1 − x y .
7. Solve (i) (x2 y − 2xy 2 )dx − (x3 − 3x2 y)dy = 0 (ii) x2 ydx − (x3 − y 3 )dy = 0.
8. Solve (i) (xy sin xy + cos xy)ydx + (xy sin xy − cos xy)xdy (ii) y(1 + xy)dx + x(1 − xy)dy = 0 (iii)
(x3 y 3 + x2 y 2 + xy + 1)ydx + (x3 y 3 − x2 y 2 − xy + 1)xdy = 0 (iv) y(x2 y 2 + 2)dx + x(2 − 2x2 y 2 )dy = 0.

9. Solve (i) (x2 + y 2 + x)dx + xydy = 0 (ii) (5xy + 4y 2 + 1)dx + (x2 + 2xy)dy = 0.
10. Solve (i) (2xy 4 ey +2xy 3 +y)dx+(x2 y 4 ey −x2 y 2 −3x)dy = 0 (ii) (xy 2 −x2 )dx+(3x2 y 2 +x2 y−2x3 +y 2 )dy =
0
Leibniz’s linear ODE
dy dy dy
p
11. Solve (i) x cos x dx + y(x sin x + cos x) = 1 (ii) (1 − x2 ) dx + 2xy = x (1 − x2 ) (iii) (x + 2y 3 ) dx =y
2 −1 2 3 dy 1
(iv) (1 + y )dx = (tan y − x)dy (v) (1 + x + xy )dy + (y + y )dy = 0 (vi) dx + y cos x = 2 sin 2x.
Bernoulli’s equation
dy dy dy dy
12. Solve (i) dx + x sin 2y = x3 cos2 y (ii) dx = ex−y (ex − ey ) (iii) x dx + y log y = xyex (iv) dx + x1 sin 2y =
2 2 2 2
x cos y (v) (x + y + 2y)dy + 2xdx = 0.
ODE of 1st order and of higher degree
13. Solve (i) p3 = ax4 (ii) 4xp2 = (3x − a)2 (iii) p2 − 7p + 12 = 0 (iv) x2 p2 + xyp − 6y 2 = 0 (v)
x2 p2 − 2xyp + 2y 2 − x2 = 0.
14. Solve (i) y = 2px + y 2 p3 (ii) y + px = x4 p2 .
Clairaut’s equation
15. Solve (i) y = px + (1 + p2 )1/2 (ii) p = log(px − y) (iii) sin px cos y = cos px sin y + p (iv) p2 (x2 − a2 ) −
2pxy + y 2 − b2 = 0 (v) p2 x(x − 2) + p(2y − 2xy − x + 2) + y 2 + y = 0.
∗
ODE of 2nd and higher order with constant co-efficients
16. Solve (i) (D3 + 6D2 + 11D + 6)y = 0 (ii) (D4 − 5D2 + 4)y = 0 (iii) (D4 + 8D2 + 16)y = 0 (iv)
(D2 + D + 1)2 (D − 2)y = 0.
17. Solve (i) (D2 +a2 )y = sec ax (ii) (D2 −3D+2)y = e5x (iii) (D2 +4D+3)y = e−3x (iv) (D2 −3D+2)y =
sin 3x (v) (D2 + a2 )y = cos ax (vi) (D4 − m4 )y = sin mx (vii) (D2 − 3D + 2)y = sin 2x + xe3x .
18. Solve (i) (D4 −D2 )y = 2 (ii) (D2 +D)y = x2 +2x+4 (iii) (D3 +3D2 +2D)y = x2 (iv) (D3 −D2 −6D)y =
x2 + 1.
19. Solve (i) (D2 − 2D + 1)y = x2 e3x (ii) (D − 1)2 y = ex sec2 x tan x (iii) (D2 − 4D + 4)y = e2x sin 2x.
20. Solve (i) (D2 + 9)y = x sin x (ii) (D2 + 2D + 1)y = x cos x.

Algebraic structures
21. What is an algebraic structure? What is a (i) binary operation (ii) n-ary operation? Give examples.
22. Define: Group, Abelian group. Give examples.
23. Prove that the identity element of a group and inverse of any element in a group is unique.
24. Let G be a group with the property that for any x, y, z in the group, xy = zx implies y = z. Prove that
G is abelian.
25. While constructing a group under multiplication modulo 91, an integer was left out and the list appeared
as 1, 9, 16, 22, 53, 74, 79, 81. Which integer was left out?
26. Prove that if every element of a group is its own inverse, then the group is abelian.
27. Prove that if the square of every element in a group is identity, then the group is abelian.
28. Define: subgroup, permutation group, symmetric group Sn . Give examples.
29. Define: Ring, Integral Domain, Field. Give examples.
30. Show that the set {0, 2, 4} under addition and multiplication modulo 6 is a ring. Find the unity element
of this ring.
31. Show that for fixed nonzero elements a and b in a ring, the equation ax = b can have more than one
solution. How does it compare with a group?
32. Prove that cancellation law holds in an integral domain.
33. Prove that every finite integral domain is a field. Give example of an integral domain which is not a
field.
34. Prove that Zp is a field if p is a prime.
35. What is a zero divisor? Give example.
36. Prove that the set of all n × n matrices forms a group under addition of matrices.
37. Find the number of elements in the symmetric group Sn .