#### The number of arbitrary constants in the particular solution of a differential equation of m order is ___________, where m is an integer.

By definition, the solution that is free from arbitrary constants i.e., the solution obtained from the general solution by giving particular values to the arbitrary constants is called a particular solution of the differential equation. Therefore, the number of arbitrary constants in the particular solution of a differential equation is zero. Hence, the number of arbitrary constants in the particular solution of a differential equation of m order is 0, where m is an integer.

#### The highest order derivative of the dependent variable with respect to the independent variable involved in the given differential equation is called ___________ of the differential equation.

By definition, order of a differential equation is defined as the order of the highest order derivative of the dependent variable with respect to the independent variable involved in the given differential equation.

#### An equation involving derivatives of the dependent variable with respect to independent variables is called a/ an _______________________.

By definition, an equation involving derivative (derivatives) of the dependent variable with respect to independent variable (variables) is called a differential equation.

#### A solution of differential equation which contains arbitrary constants is called the _______ of the differential equation

By definition, a solution of a differential equation which contains arbitrary constants is called the general solution of the differential equation.

#### A differential equation involving derivatives of the dependent variable with respect to only one independent variable is called a/an _______________.

By definition, a differential equation involving derivatives of the dependent variable with respect to only one independent variable is called an ordinary differential equation.

#### If a differential equation is of degree 3 and order 2, then the number of arbitrary constants in the general solution of the differential equation is _________.

In the general solution of a differential equation, the number of arbitrary constants in the general solution of a differential equation is equal to the order of the differential equation. Here, given that the order of the differential equation is 2, the number of arbitrary constants in the general solution of the differential equation is 2. Hence, if a differential equation is of degree 3 and order 2, then the number of arbitrary constants in the general solution of the differential equation is 2.

#### A function y = a sin x – b cos x, where a ∈ R, is a solution of the differential equation ________________.

The given function is y = a sin x – b cos x ------(1) Differentiating both sides of equation (1) successively with respect to x, we get y' = a cos x + b sin x y'' = -a sin x + b cos x ⇒ y'' = -(a sin x - b cos x) ⇒ = -y Therefore, y'' + y = 0 Hence, a function y = a sin x – b cos x, where a ∈ R is a solution of the differential equation y''+ y = 0.

#### The degree of the differential equation 4y'- y/y' = sin x is _________.

By definition, the degree of a differential equation is the power (positive integral index) of the highest derivative involved in the given differential equation. The given differential equation is 4y'- y/y' = sin x. ⇒ 4(y')^{2}- (sin x)y' - y = 0 Therefore, power (positive integral index) of the highest derivative involved in the given differential equation is 2. Hence, the degree of the differential equation 4y'- y/y' = sin x is 2.

#### The order of the differential equation 5y + 3(y'''')^{3}- 6y' = 0 is _________.

By definition, Order of a differential equation is defined as the order of the highest derivative of the dependent variable with respect to the independent variable involved in the given differential equation. In the given differential equation 5y + 3(y'''')^{3}- 6y' = 0, the highest derivative of the dependent variable with respect to the independent variable is 4 i.e. y'''' Hence, the order of the differential equation 5y + 3(y'''')^{3}- 6y' = 0 is 4.

#### A solution of differential equation which is free from arbitrary constants is called a ___________ of the differential equation.

By definition, the solution that is free from arbitrary constants i.e., the solution obtained from the general solution by giving particular values to the arbitrary constants, is called a particular solution of the differential equation.

#### The degree of the differential equation y'''' + y''' + y'' + y' + y = 0 is __________.

By definition, degree of a differential equation is the power (positive integral index) of the highest derivative involved in the given differential equation. The given differential equation is y'''' + y''' + y'' + y' + y = 0. The power (positive integral index) of the highest derivative (here is y'''') involved in the given differential equation is 1. Hence, the degree of the differential equation y'''' + y''' + y'' + y' + y = 0 is 1.

#### The number of arbitrary constants in the general solution of a differential equation of k order is ________, where k is a whole number.

In the general solution of a differential equation, the number of arbitrary constants in the general solution of a differential equation is equal to the order of the differential equation. Here, given that the order of the differential equation is k, the number of arbitrary constants in the general solution of the differential equation is k. Hence, the number of arbitrary constants in the general solution of a differential equation of k order is k, where k is a whole number.

#### The order of the differential equation (s')^{3} - 5(s'')^{2}- 7 = 0 is ___________

By definition, order of a differential equation is the order of the highest derivative of the dependent variable with respect to the independent variable involved in the given differential equation. In the given differential equation (s')^{3} - 5(s'')^{2}- 7 = 0, the highest derivative of the dependent variable with respect to the independent variable is 2, i.e. s''. Hence, the order of the differential equation (s')^{3} - 5(s'')^{2}- 7 = 0 is 2.

#### In a polynomial equation of derivatives, ___________ of a differential equation is the power (positive integral index) of the highest derivative term involved in the given differential equation.

By definition, in a polynomial equation in derivatives, the degree of a differential equation is the power (positive integral index) of the highest derivative term involved in the given differential equation.

#### A differential equation involving derivatives with respect to more than one independent variable is called a/an ___________________.

By definition, a differential equation involving derivatives with respect to more than one independent variable is called a partial differential equation.