Choose correct answers for the following:

1 (a) (i)
The general solution of the equation p

(a) (y-2x-c)(y-3x-c)=0

(b) (y+2x-c)(y+3x-c)=0

(c) (y-2x-c)(y+3x-c)=0

(d) (y-x-c)(y+x-c)=0

^{2}-5p+6=0 is:(a) (y-2x-c)(y-3x-c)=0

(b) (y+2x-c)(y+3x-c)=0

(c) (y-2x-c)(y+3x-c)=0

(d) (y-x-c)(y+x-c)=0

1 M

1 (a) (ii)
If a differentiable equation is solvable for y then it is of the form:

(a) x=f(y,p)

(b) y=f(x,p)

(c) y=f(x

(d) x=f(y

(a) x=f(y,p)

(b) y=f(x,p)

(c) y=f(x

^{2},py)(d) x=f(y

^{2},p)
1 M

1 (a) (iii)
The differentiable equation of the form y=px+f(p) whose general solution is y=cx+f(c) is known as?

(a) Glairauts equation

(b) Cauchys equation

(c) Lagranges equation

(d) None of these

(a) Glairauts equation

(b) Cauchys equation

(c) Lagranges equation

(d) None of these

1 M

1 (a) (iv)
The singular solution of the equation y=px-logp is:

(a) y=1-logx

(b) y=1-log(1/x)

(c) y=logx-2x

(d) None of these

(a) y=1-logx

(b) y=1-log(1/x)

(c) y=logx-2x

(d) None of these

1 M

1 (b)
Solve the equation p

^{2}+ p(x+y) + xy = 0
5 M

1 (c)
Solve the equation xp

^{2}- 2yp + ax = 0
5 M

1 (d)
Obtain the general solution and singular solution of the equation sinpx·cosy = cospx·siny+p

6 M

Choose correct answers for the following:

2 (a) (i)
The homogeneous linear differential equation whose auxiliary equation has roots 1,1,-2 is:

(a) D

(b) D

(c) (D+1)

(d) D

(a) D

^{3}+ 3D^{2}+ D + 1 = 0(b) D

^{3}- 3D + 2 = 0(c) (D+1)

^{2}(D+2) = 0(d) D

^{3}+ 3D + 2 = 0
1 M

2 (a) (ii)
The complementary function for the differential equation (D

(a) c

(b) c

(c) (c

(d) (c

^{2}+2D+1)y = 2x+x^{2}is:(a) c

_{1}e^{-x }+ x^{2}c_{2}e^{-x}(b) c

_{1}e^{x }+c_{2}e^{-x }(c) (c

_{1}+c_{2})e^{x}(d) (c

_{1}+c_{2})e^{-x}
1 M

2 (a) (iii)
The particular integral of (D

(a) (-x/2a)sinax

(b) (x/2a)cosax

(c) (-x/2a)cosax

(d) (x/2a)sinax

^{2}+a^{2})y = cosax is:(a) (-x/2a)sinax

(b) (x/2a)cosax

(c) (-x/2a)cosax

(d) (x/2a)sinax

1 M

2 (a) (iv)
The general solution of an n

(a) at most n constants

(b) exactly n independent constants

(c) atleast n independent constants

(d) more than n constants

^{th}order linear differential equation contains:(a) at most n constants

(b) exactly n independent constants

(c) atleast n independent constants

(d) more than n constants

1 M

2 (b)
Solve: y'' - 2y' + y = xe

^{x}sinx
5 M

2 (c)
Solve: d

^{2}y/dx^{2}- 4dy/dx + 4y = e^{2x }+ cosx + 4
6 M

2 (d)
Solve: dx/dt = 2x-3y, dy/dt = y-2x given x(0) =8 and y(0) = 3

6 M

Choose correct answers for the following:

3 (a) (i)
By the method of variation of parameters,the value of W is called:

(a) Demorgans function

(b) Eulers function

(c) Wronskian function

(d) None of these

(a) Demorgans function

(b) Eulers function

(c) Wronskian function

(d) None of these

1 M

3 (a) (ii)
The differential equation of the form a

(a) Simultaneous equation

(b) Legendres equation

(c) Cauchys equation

(d) Eulers equation

_{0}(ax+b)^{2}y''+a_{1}(ax+b)y'+a_{2}y = φ(x) is called:(a) Simultaneous equation

(b) Legendres equation

(c) Cauchys equation

(d) Eulers equation

1 M

3 (a) (iii)
The equation x

(a) (D

(b) D

(c) D

(d) none of these

^{3}d^{3}y/dx^{3}+ 3x^{2}d^{2}y/dx^{2}+ x dy/dx = x^{3}logx by putting x=e^{t}with D=d/dt reduces to?(a) (D

^{3}+D^{2}+D)y = 0(b) D

^{3}y = 0(c) D

^{3}y = te^{3t}(d) none of these

1 M

3 (a) (iv)
To find the series solution for the equation 4xd

^{2}y/dx^{2}+ 2dy/dx + y = 0,we assume the solution as:
1 M

3 (b)
Using the variation of parameters method,solve the equation y'' - 2y' + y = e

^{x}/x
4 M

3 (c)
Solve the equation : x

^{2}y'' - xy' + 2y = xsin(logx)
6 M

3 (d)
Obtain the Frobenius type series solution of the equation xd

^{2}y/dx^{2}+ y = 0
6 M

Choose correct answers for the following:

4 (a) (i)
The partial differential equation obtained by eliminating arbitrary constants from the relation Z = (x-a

(a) p

(b) p

(c) p + q = z

(d) p - q = 2z

^{2}) + (y-b^{2}) is:(a) p

^{2}+ q^{2}= 4z(b) p

^{2}- q^{2}= 4z(c) p + q = z

(d) p - q = 2z

1 M

4 (a) (ii)
The auxiliary equations of Lagranges linear equation Pp + Qq = R = are:

(a) dx/p = dy/q = dz/R

(b) dx/P = dy/Q = dz/R

(c) dx/x = dy/y = dz/z

(d) dx/x + dy/y - dz/z = 0

(a) dx/p = dy/q = dz/R

(b) dx/P = dy/Q = dz/R

(c) dx/x = dy/y = dz/z

(d) dx/x + dy/y - dz/z = 0

1 M

4 (a) (iii)
General solution of the equation ∂

(a) (1⁄6) x

(b) (1⁄6) x

(c) (1⁄6) x

(d) None of these

^{2}z/(∂x∂y) = x^{2}y is:(a) (1⁄6) x

^{3}y^{2}+ f(y) + g(x)(b) (1⁄6) x

^{2}y^{2}+ f(y)(c) (1⁄6) x

^{3}y^{3}(d) None of these

1 M

4 (a) (iv)
By the method of separation of variables,we seek a solution in the form:

(a) X = X(x)Y(y)

(b) Z = X+Y

(c) Z = X

(d) Z =X/Y

(a) X = X(x)Y(y)

(b) Z = X+Y

(c) Z = X

^{2}Y^{2}(d) Z =X/Y

1 M

4 (b)
Form a partial differential equation from the relation Z = f(y) + φ(x+y)

5 M

4 (c)
Solve the equation (x

^{2}- y^{2}- z^{2})p + 2xyq = 2xz
5 M

4 (d)
Use the method of separation of variables to solve ∂u/∂x = 2∂u/∂t + u; given that u(x,0) = 6e

^{-3x}
6 M

Choose correct answers for the following:

5 (a) (i)
∫

(a) 1⁄2

(b) -1⁄2

(c) 1⁄4

(d) 2⁄5

_{0}^{1}∫_{0}^{x2}e^{y/x }dydx is equal to:(a) 1⁄2

(b) -1⁄2

(c) 1⁄4

(d) 2⁄5

1 M

5 (a) (ii)
The integral ∫

_{0}^{∞}∫_{0}^{∞}e^{-(x2+y2)}dxdy by changing to polar form becomes:
1 M

5 (a) (iii)
β (3,1⁄2) is equal to:

(a) 16⁄11

(b) 16⁄15

(c) 15⁄16

(d) 2π/3

(a) 16⁄11

(b) 16⁄15

(c) 15⁄16

(d) 2π/3

1 M

5 (a) (iv)
The integral 2∫

(a) Γ(3⁄2)

(b) Γ(n+1)

(c) Γ(-1⁄2)

(d) Γ(1⁄2)

_{0}^{∞}e^{-x2}dx is:(a) Γ(3⁄2)

(b) Γ(n+1)

(c) Γ(-1⁄2)

(d) Γ(1⁄2)

1 M

5 (b)
Evaluate by changing the order of integration ∫

_{0}^{a}∫_{0}^{2√(xa)}x^{2}dydx; a>0
4 M

5 (c)
Evaluate the integral ∫

_{0}^{1}∫_{0}^{√(1-x2)}∫_{0}^{√(1-x2-y2)}xyz dzdydx
6 M

5 (d)
Prove that ∫

_{0}^{∞}xe^{-x8}dx × ∫_{0}^{∞}x^{2}e^{-x4}dx = π/(16√2)
6 M

Choose correct answers for the following:

6 (a) (i)
If f = (5xy-6x

(a) 35

(b) -35

(c) 3x+4y

(d) None of these

^{2})i + (2y-4x)j then ∫_{C}f·dr where C is the curve y=x^{3}from the points (1,1) to (2,8) is:(a) 35

(b) -35

(c) 3x+4y

(d) None of these

1 M

6 (a) (ii)
In Green's theorem in the plane ∫

_{C}(Mdx+Ndy) = ?
1 M

6 (a) (iii)
If ∫

(a) Rational

(b) Irrotational

(c) Solenoidal

(d) Rotational

_{C}f·dr^{→}= 0 then f is called:(a) Rational

(b) Irrotational

(c) Solenoidal

(d) Rotational

1 M

6 (a) (iv)
If all the surfaces are closed in a region containing volume V then the following theorem is applicable:

(a) Stokes theorem

(b) Greens theorem

(c) Gauss divergence theorem

(d) none of these

(a) Stokes theorem

(b) Greens theorem

(c) Gauss divergence theorem

(d) none of these

1 M

6 (b)
If f = (2x

^{2}-3z)i - 2xyj - 4xk. Evaluate∫_{v}curl f·dv where v is the volume of the region bounded by the planes x=0, y=0, z=0 and 2x+2y+z = 4
4 M

6 (c)
Verify Green's theorem for ∫

_{c}(3x^{2}- 8y^{2})dx + (4y-6xy)dy where c is the triangle formed by x=0, y=0 and x+y=1
6 M

6 (d)
Verify the Stokes theorem for f = -y

^{3}i^{^}+ x^{3}j^{^ }where s is the circular disc x^{2}+ y^{2}≤ 1, z=0
6 M

Choose correct answers for the following:

7 (a) (i)
The Laplace transform of f(t)/t when L[ f(t) ] = F(s) is:

(a) ∫

(b) ∫

(c) ∫

(d) ∫

(a) ∫

_{0}^{∞}F(s) ds(b) ∫

_{s}^{∞}F(s) ds(c) ∫

_{0}^{∞}F(s-a) ds(d) ∫

_{0}^{∞}F(s+a) ds
1 M

7 (a) (ii)
L[t

(a) 3!/(s-2)

(b) 3!/(s+2)

(c) 3/(s-2)

(d) 3/(s-2)

^{3}e^{2t}] = ?(a) 3!/(s-2)

^{4}(b) 3!/(s+2)

^{4}(c) 3/(s-2)

^{4}(d) 3/(s-2)

1 M

7 (a) (iii)
L [f(t-a)H(t-a)] is equal to:

(a) e

(b) e

(c) e

(d) L[f(t)]/se

(a) e

^{-as}L[f(t)](b) e

^{as}L[f(t)](c) e

^{-as}/s(d) L[f(t)]/se

^{-as}
1 M

7 (a) (iv)
L[δ(t)] is equal to:

(a) 0

(b) -1

(c) e

(d) L

(a) 0

(b) -1

(c) e

^{-as}(d) L

1 M

7 (b)
Evaluate L[sint sin2t sin3t]

4 M

7 (c)
A periodic function of 2π/ω is defined by:

f(t) = Esinωt for 0 ≤ t ≤ π/ω

f(t) = 0 for π/ω ≤ 1 ≤ 2π/ω

Find L[f(t)].

f(t) = Esinωt for 0 ≤ t ≤ π/ω

f(t) = 0 for π/ω ≤ 1 ≤ 2π/ω

Find L[f(t)].

6 M

7 (d)
Express in terms of unit step function and hence find L[f(t)]

f(t) = 2t , 0 < t ≤ π

f(t) = 1, t > π

f(t) = 2t , 0 < t ≤ π

f(t) = 1, t > π

6 M

Choose correct answers for the following:

8 (a) (i)
L

(a) ∫

(b) ∫

(c) ∫

(d) ∫

^{-1}[F(s)/s] is equal to:(a) ∫

_{0}^{1}f(t)·dt(b) ∫

_{0}^{∞}f(t)·dt(c) ∫

_{0}^{∞}f(t-a)·dt(d) ∫

_{0}^{t}f(t-a)·d
1 M

8 (a) (ii)
L

(a) e

(b) 1⁄2e

(c) 1⁄2e

(d) e

^{-1}[1/(s^{2}+2s+5)] is equal to:(a) e

^{t}sin2t(b) 1⁄2e

^{-t}sin2t(c) 1⁄2e

^{t}cos2t(d) e

^{2}tcos2t
1 M

8 (a) (iii)
f(t)×g(t) is defined by:

(a) ∫

(b) ∫

(c) ∫

(d) ∫

(a) ∫

_{0}^{t}f(t-u)g(u)du(b) ∫

_{0}^{∞}f(t)dt(c) ∫

_{0}^{t}f(t)g(t)du(d) ∫

_{0}^{t}f(u)g(u)du
1 M

8 (a) (iv)
L

(a) cosat

(b) secat

(c) sinat

(d) (1/a)sinat

^{-1}[1/(s^{2}+a^{2})] is:(a) cosat

(b) secat

(c) sinat

(d) (1/a)sinat

1 M

8 (b)
Find L

^{-1}{(2s-1)/(s^{2}+2s+17)}
4 M

8 (c)
By employing the convolution theorem evaluate L

^{-1}{s/(s^{2}+a^{2})^{2}}
6 M

8 (d)
Solve the initial value problem y'' - 3y' + 2y = 4t + e

^{3t}; y(0)=1, y'(0) = -1 using Laplace transforms
6 M

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