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Part II CODE: XXXX

(1) Let y2 = 4ax be a family of parabolas. The orthogonal trajectories to thefamily are:(A) Hyperbolas with varying eccentricities.(B) Hyperbolas with a common eccentricity.(C) Ellipses with varying eccentricities.

(D) Ellipses with a common eccentricity.√

(2) The differential (cosx+ y sin x)dx+ x sin xdy has an Integrating Factor(A) which is of the form µ(x) and has no I.F. of the form µ(y).(B) which is of the form µ(y) and has no I.F. of the form µ(x).(C) which is of the form µ(x) and also an I.F. of the form µ(y).

(D) has no I.F. of the form µ(x) or µ(y).√

(3) The function f(x, y) = sin√x+ cos

√y in the square {0 ≤ x ≤ 1, 0 ≤ y ≤ 1}

is(A) Lipshitz w.r.t. x but not w.r.t y

(B) Lipshitz w.r.t. y but not w.r.t x√

(C) Lipshitz w.r.t. x and also w.r.t y(D) Lipshitz neither w.r.t. x nor w.r.t y

(4) Which of the following is exact in R2 \ {(0, 0)}(A)

ydx− xdy

x2 + y2

(B)ydx+ xdy

x2 + y2

(C)xdx+ ydy

x2 + y2√

(D)xdx− ydy

x2 + y2

(5) The number of distinct explicit solutions of the differential equation 6y�+y3+2

x√x= 0 which are of the form

C√xis

(A) 0.(B) 1.

(C) 2.√

(D) 3.(6) A successful choice for a particular solution of y�� − 2y� + 5y = x2ex sin 2x will

be of the form(A) xex[(Ax2 + Bx+ C) cos 2x+ (Px2 +Qx+R) sin 2x].

√

(B) xex(Ax2 + Bx+ C)(P cos 2x+Q sin 2x).(C) xe2x[(Ax2 + Bx+ C) cosx+ (Px2 +Qx+R) sin x].(D) x2e2x(Ax+B)(P cos x+Q sin x).

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Part II CODE: XXXX

(1) Let y2 = 4ax be a family of parabolas. The orthogonal trajectories to thefamily are:(A) Hyperbolas with varying eccentricities.(B) Hyperbolas with a common eccentricity.(C) Ellipses with varying eccentricities.

(D) Ellipses with a common eccentricity.√

(2) The differential (cosx+ y sin x)dx+ x sin xdy has an Integrating Factor(A) which is of the form µ(x) and has no I.F. of the form µ(y).(B) which is of the form µ(y) and has no I.F. of the form µ(x).(C) which is of the form µ(x) and also an I.F. of the form µ(y).

(D) has no I.F. of the form µ(x) or µ(y).√

(3) The function f(x, y) = sin√x+ cos

√y in the square {0 ≤ x ≤ 1, 0 ≤ y ≤ 1}

is(A) Lipshitz w.r.t. x but not w.r.t y

(B) Lipshitz w.r.t. y but not w.r.t x√

(C) Lipshitz w.r.t. x and also w.r.t y(D) Lipshitz neither w.r.t. x nor w.r.t y

(4) Which of the following is exact in R2 \ {(0, 0)}(A)

ydx− xdy

x2 + y2

(B)ydx+ xdy

x2 + y2

(C)xdx+ ydy

x2 + y2√

(D)xdx− ydy

x2 + y2

(5) The number of distinct explicit solutions of the differential equation 6y�+y3+2

x√x= 0 which are of the form

C√xis

(A) 0.(B) 1.

(C) 2.√

(D) 3.(6) A successful choice for a particular solution of y�� − 2y� + 5y = x2ex sin 2x will

be of the form(A) xex[(Ax2 + Bx+ C) cos 2x+ (Px2 +Qx+R) sin 2x].

√

(B) xex(Ax2 + Bx+ C)(P cos 2x+Q sin 2x).(C) xe2x[(Ax2 + Bx+ C) cosx+ (Px2 +Qx+R) sin x].(D) x2e2x(Ax+B)(P cos x+Q sin x).

5

(7) The inhomogeneous Cauchy-Euler equation

x2y�� + 2xy� +y

4=

a√x+ b cos(c ln x) + xekx

can be solved by the method of undetermined cefficients(A) For arbitrary values of a, b, c and k.

(B) For arbitrary values of a, b and c but when k = 0.√

(C) For arbitrary values of b, c and k but when a = 0.(D) For arbitrary values of a, b and k but when c = 0.

(8) Let xex and x2ex be two linearly independent solutions of a second order lineardifferential equation LY ≡ y�� + p(x)y� + q(x)y = 0. A particular solution ofLy = ex is(A) −x2ex ln x.(B) −x2ex + x ln x.

(C) x2ex ln x.√

(D) −xex ln x.(9) The solution of the Initial Value problem

dy

dx=

y2 − xy

x2 + xy, y(1) = 1 is

(A) ln xy +y

x= 1.

√

(B) ln xy +y

x= 0.

(C) ln xy − x

y= 1.

(D) ln xy − x

y= 0.

(10) The general solution of the equation 4y − y�� = 4x2 is

(A) A cosh 2x+ B sinh 2x− x2 − 1

2.

(B) Ae2x + Be−2x − x2 +1

2.

(C) Ae2x + Be−2x + x2 − 1

2.

(D) A cosh 2x+ B sinh 2x+ x2 +1

2.

√

(11) The Laplace transform of f(t) = t cosωt is

(A)s2 − ω2

(s2 + ω2)2.

√

(B)ω2 − s2

(s2 + ω2)2.

(C)s2 + ω2

(s2 − ω2)2.

(D)2sω

(s2 + ω2)2.

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(7) The inhomogeneous Cauchy-Euler equation

x2y�� + 2xy� +y

4=

a√x+ b cos(c ln x) + xekx

can be solved by the method of undetermined cefficients(A) For arbitrary values of a, b, c and k.

(B) For arbitrary values of a, b and c but when k = 0.√

(C) For arbitrary values of b, c and k but when a = 0.(D) For arbitrary values of a, b and k but when c = 0.

(8) Let xex and x2ex be two linearly independent solutions of a second order lineardifferential equation LY ≡ y�� + p(x)y� + q(x)y = 0. A particular solution ofLy = ex is(A) −x2ex ln x.(B) −x2ex + x ln x.

(C) x2ex ln x.√

(D) −xex ln x.(9) The solution of the Initial Value problem

dy

dx=

y2 − xy

x2 + xy, y(1) = 1 is

(A) ln xy +y

x= 1.

√

(B) ln xy +y

x= 0.

(C) ln xy − x

y= 1.

(D) ln xy − x

y= 0.

(10) The general solution of the equation 4y − y�� = 4x2 is

(A) A cosh 2x+ B sinh 2x− x2 − 1

2.

(B) Ae2x + Be−2x − x2 +1

2.

(C) Ae2x + Be−2x + x2 − 1

2.

(D) A cosh 2x+ B sinh 2x+ x2 +1

2.

√

(11) The Laplace transform of f(t) = t cosωt is

(A)s2 − ω2

(s2 + ω2)2.

√

(B)ω2 − s2

(s2 + ω2)2.

(C)s2 + ω2

(s2 − ω2)2.

(D)2sω

(s2 + ω2)2.

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(12) Let 0 < a < b and F (s) = lns+ b

s+ a= (L f)(s). Then f itself is a Laplace

transform L g, where g is(A) a periodic function.(B) a nondecreasing piecewise constant function.

(C) a piecewise constant function.√

(D) a nonincreasing piecewise constant function.

(13) The solution of the IVP y = sinϕ−� t

0

y(t− τ)dτ, y(0) = cosϕ is

(A) cos(t+ ϕ).(B) sin(t− ϕ).(C) sin(t+ ϕ).

(D) cos(t− ϕ).√

(14) Let J0(t) be the Bessel’s function defined by J0(t) =∞�

k=0

(−1)k�

tk

2kk!

�2

. The

Laplace transform (L J0)(s) is

(A) (s2 + 1)−12 .

√

(B) (s2 + 1)−32 .

(C) (s2 − 1)−12 .

(D) (s2 − 1)−32 .

(15) Let U(t) =∞�

k=0

uk(t), where uk(t) denotes the shifted Heaviside function

u(t− k). Then The Laplace transform of U(t)− t is

(A)

es� 1

0

te−stdt

1− e−s.

(B)

e−s

� 1

0

testdt

1− e−s.

√

(C)

es� 1

0

te−stdt

es − 1.

(D)

e−s

� 1

0

te−stdt

es − 1.

ENDAll the Best

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(12) Let 0 < a < b and F (s) = lns+ b

s+ a= (L f)(s). Then f itself is a Laplace

transform L g, where g is(A) a periodic function.(B) a nondecreasing piecewise constant function.

(C) a piecewise constant function.√

(D) a nonincreasing piecewise constant function.

(13) The solution of the IVP y = sinϕ−� t

0

y(t− τ)dτ, y(0) = cosϕ is

(A) cos(t+ ϕ).(B) sin(t− ϕ).(C) sin(t+ ϕ).

(D) cos(t− ϕ).√

(14) Let J0(t) be the Bessel’s function defined by J0(t) =∞�

k=0

(−1)k�

tk

2kk!

�2

. The

Laplace transform (L J0)(s) is

(A) (s2 + 1)−12 .

√

(B) (s2 + 1)−32 .

(C) (s2 − 1)−12 .

(D) (s2 − 1)−32 .

(15) Let U(t) =∞�

k=0

uk(t), where uk(t) denotes the shifted Heaviside function

u(t− k). Then The Laplace transform of U(t)− t is

(A)

es� 1

0

te−stdt

1− e−s.

(B)

e−s

� 1

0

testdt

1− e−s.

√

(C)

es� 1

0

te−stdt

es − 1.

(D)

e−s

� 1

0

te−stdt

es − 1.

ENDAll the Best