The order and degree of the differential equation \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +{ \left( \frac { dy }{ dx } \right) }^{ 1/3 }+{ x }^{ 1/4 }=0\) are respectively

(a)

2, 3

(b)

3, 3

(c)

2, 6

(d)

2, 4

The differential equation representing the family of curves y = Acos(x + B), where A and B are parameters,is

(a)

\(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } -y=0\)

(b)

\(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } }+y=0\)

(c)

\(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } }=0\)

(d)

\(\frac { { d }^{ 2 }x }{ { dy }^{ 2 } }=0\)

The order and degree of the differential equation \(\sqrt { sinx } (dx+dy)=\sqrt { cos x } (dx-dy)\) is

(a)

1, 2

(b)

2, 2

(c)

1, 1

(d)

2, 1

The order of the differential equation of all circles with centre at (h, k ) and radius ‘a’ is

(a)

2

(b)

3

(c)

4

(d)

1

The differential equation of the family of curves y = Ae^x + Be^−x, where A and B are arbitrary constants is

(a)

\(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +y=0\)

(b)

\(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } -y=0\)

(c)

\(\frac { { d }y }{ { dx } } +y=0\)

(d)

\(\frac { { d }y }{ { dx } } -y=0\)

The general solution of the differential equation \(\frac { dy }{ dx } =\frac { y }{ x } \) is

(a)

xy = k

(b)

y = k log x

(c)

y = kx

(d)

log y = kx

The solution of the differential equation \(2x\frac{dy}{dx}-y=3\) represents

(a)

straight lines

(b)

circles

(c)

parabola

(d)

ellipse

The solution of \(\frac{d y}{d x}+p(x) y=0\) is

(a)

\(y={ ce }^{ \int { pdx } }\)

(b)

\(y={ ce }^{ -\int { pdx } }\)

(c)

\(x={ ce }^{ -\int { pdy } }\)

(d)

\(x={ce }^{ \int { pdy } }\)

The integrating factor of the differential equation \(\frac { dy }{ dx } +y=\frac { 1+y }{ \lambda } \) is

(a)

\(\frac { x }{ { e }^{ \lambda } } \)

(b)

\(\frac { { e }^{ \lambda} }{ x } \)

(c)

\({ \lambda e }^{ x }\)

(d)

e^x

The integrating factor of the differential equation \(\frac{d y}{d x}+P(x) y=Q(x)\) is x, then P(x)

(a)

x

(b)

\(\frac { { x }^{ 2 } }{ 2 } \)

(c)

\(\frac{1}{x}\)

(d)

\(\frac{1}{x^2}\)

The degree of the differential equation \(y(x)=1+\frac { dy }{ dx } +\frac { 1 }{ 1.2 } { \left( \frac { dy }{ dx } \right) }^{ 2 }+\frac { 1 }{ 1.2.3 } { \left( \frac { dy }{ dx } \right) }^{ 3 }+....\) is

(a)

2

(b)

3

(c)

1

(d)

4

If p and q are the order and degree of the differential equation \(y=\frac { dy }{ dx } +{ x }^{ 3 }\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) +xy=cosx,\) When

(a)

p < q

(b)

p = q

(c)

p > q

(d)

p exists and q does not exist

The solution of the differential equation \(\frac { dy }{ dx } +\frac { 1 }{ \sqrt { 1-{ x }^{ 2 } } } =0\) is

(a)

y + sin^-1 x = c

(b)

x + sin^-1 y = 0

(c)

y²+ 2 sin^-1 x = c

(d)

x²+ 2 sin^-1y= c

The solution of the differential equation \(\frac { dy }{ dx } =2xy\) is

(a)

y = Ce^_x2

(b)

y = 2x² + C

(c)

y = Ce^_−x2 + C

(d)

y = x² + C

The general solution of the differential equation \(\log \left(\frac{d y}{d x}\right)=x+y\) is

(a)

e^x + e^y = C

(b)

e^x + e^-y = C

(c)

e^-x + e^y = C

(d)

e^-x + e^-y = C

The solution of \(\frac { dy }{ dx } ={ 2 }^{ y-x }\) is

(a)

2^x + 2^y = C

(b)

2^x - 2^y = C

(c)

\(\frac { 1 }{ { 2 }^{ x } } -\frac { 1 }{ { 2 }^{ y } } =C\)

(d)

x + y = C

The solution of the differential equation \(\frac{d y}{d x}=\frac{y}{x}+\frac{\phi\left(\frac{y}{x}\right)}{\phi^{\prime}\left(\frac{y}{x}\right)}\) is

(a)

\(x\phi \left( \frac { y }{ x } \right) =k\)

(b)

\(\phi \left( \frac { y }{ x } \right) =kx\)

(c)

\(y\phi \left( \frac { y }{ x } \right) =k\)

(d)

\(\phi \left( \frac { y }{ x } \right) =ky\)

If sin x is the integrating factor of the linear differential equation \(\frac { dy }{ dx } +Py=Q,\) then P is

(a)

log sin x

(b)

cos x

(c)

tan x

(d)

cot x

The number of arbitrary constants in the general solutions of order n and n +1 are respectively

(a)

n-1,n

(b)

n,n+1

(c)

n+1,n+2

(d)

n+1,n

The number of arbitrary constants in the particular solution of a differential equation of third order is

(a)

3

(b)

2

(c)

1

(d)

0

Integrating factor of the differential equation \(\frac{d y}{d x}=\frac{x+y+1}{x+1}\) is 

(a)

\(\frac{1}{x+1}\)

(b)

x+1

(c)

\(\frac { 1 }{ \sqrt { x+1 } } \)

(d)

\({ \sqrt { x+1 } } \)

The population P in any year t is such that the rate of increase in the population is proportional to the population. Then

(a)

P=Ce^kt

(b)

P=Ce^-kt

(c)

P=Ckt

(d)

P=C

P is the amount of certain substance left in after time t. If the rate of evaporation of the substance is proportional to the amount remaining, then

(a)

P = Ce^kt

(b)

P = Ce^-kt

(c)

P = Ckt

(d)

Pt = C

If the solution of the differential equation \(\frac{d y}{d x}=\frac{a x+3}{2 y+f}\)represents a circle, then the value of a is

(a)

2

(b)

-2

(c)

1

(d)

-1

The slope at any point of a curve y = f (x) is given by \(\frac{dy}{dx}=3x^2\) and it passes through (-1, 1). Then the equation of the curve is

(a)

y = x³ + 2

(b)

y = 3x² + 4

(c)

y = 3x⁴ + 4

(d)

y = 3x² + 5