The general solution of a first-order differential equation dy/dx = f(x) is given by ___?

∫ f(x) dx + C

True or False: The order of a differential equation refers to the highest exponent of the dependent variable in the equation.

False. The order of a differential equation is defined by the highest derivative present in the equation, not the exponent of the dependent variable.

A differential equation of the form dy/dx + P(x)y = Q(x) is called a ___?

Linear first-order differential equation.

Fill in the blank: A differential equation involving partial derivatives with respect to more than one independent variable is called a ___?

Partial Differential Equation.

What is the difference between the order and degree of a differential equation?

The order is the highest derivative present, while the degree is the highest power of the highest order derivative when the equation is made free of radicals and fractions.

True or False: A linear differential equation cannot contain products of the dependent variable and its derivatives.

True. A linear differential equation must only involve the dependent variable and its derivatives to the first degree.

The method of solving a differential equation by separating variables involves rearranging the equation into the form ___?

F(x)dx = g(y)dy.

For a homogeneous differential equation, the expressions f(x, y) and g(x, y) must be ___?

Homogeneous of the same degree.

What is the integrating factor for a linear differential equation of the form dy/dx + P(x)y = Q(x)?

E^(∫ P(x) dx).

Fill in the blank: The particular solution of a differential equation is obtained by assigning specific values to the ___?

Arbitrary constants.

True or False: A Bernoulli's equation is of the form dy/dx + P(x)y = Qy^n, where n can be any real number.

True. Bernoulli's equation has the specified form, where n is not equal to 0 or 1.

A differential equation of the form d²y/dx² = f(x) is solved by ___?

Integrating f(x) twice.

Fill in the blank: A solution of a differential equation with arbitrary constants equal to the order of the equation is called a ___?

General solution.

What is the form of a second-order linear differential equation?

A₂(x)d²y/dx² + a₁(x)dy/dx + a₀(x)y = f(x).

Fill in the blank: The process of forming a differential equation from a given relation involves differentiation and the elimination of ___?

Arbitrary constants.