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

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