Differential Equations

Differential equations are widely used because they help describe

1. Only geometric constructions
2. Only algebraic identities
3. Physical and real-life processes involving change
4. Only matrix operations

The solution obtained after applying the given initial condition to a general solution is called

1. Complementary solution
2. Particular solution
3. Auxiliary solution
4. Singular solution

Which of the following functions satisfies the differential equation dy/dx = y?

1. y = x^2
2. y = e^x
3. y = ln x
4. y = x + 1

Which of the following is the order of the differential equation d^3y/dx^3 + (dy/dx)^2 + y = 0?

1. 1
2. 2
3. 3
4. 4

Which of the following is the particular solution of the differential equation dy/dx = 2x, if y = 10 when x = 3?

1. y = x^2 + 1
2. y = x^2 + 2
3. y = x^2 + 3
4. y = x^2 + 4

The complete solution of dy/dx = 0 is

1. y = x + C
2. y = C
3. y = x^2 + C
4. y = e^x

The differential equation dy/dx = x/y belongs to which type?

1. Variable separable differential equation
2. Linear equation with constant coefficients
3. Second-order differential equation
4. Homogeneous linear equation

The solution of dy/dx = 2x, given that y = 5 when x = 1, is

1. y = x^2 + 4
2. y = x^2 + 5
3. y = 2x^2 + 3
4. y = x^2 + 6

Which of the following represents the general solution of a differential equation?

1. A solution containing no arbitrary constant
2. A solution containing one or more arbitrary constants
3. A numerical value only
4. A matrix equation

Solve the differential equation dy/dx = 3x^2.

1. y = x^3 + C
2. y = 3x^3 + C
3. y = x^2 + C
4. y = 9x + C

The differential equation dy/dx = ky, where k is a constant, represents

1. Polynomial growth
2. Exponential growth or decay
3. Circular motion
4. Quadratic relation

The general solution of dy/dx = 4x is

1. y = 2x^2 + C
2. y = 4x^2 + C
3. y = x^2 + C
4. y = 8x + C

The degree of the differential equation (dy/dx)^2 + y = 0 is

1. 1
2. 2
3. 3
4. Not defined

The order of the differential equation d^2y/dx^2 + 3dy/dx + y = 0 is

1. 1
2. 2
3. 3
4. 0

A differential equation is an equation involving

1. Only algebraic expressions
2. A function and one or more of its derivatives
3. Only constants
4. Only matrices