Differential Differential Equations Equ ations Overview: ● ● ● An equation involving derivative(s) of the dependent variable with respect to independent variable(s) is known as a differential equation. A differential equation involving derivatives of the dependent variable with respect to only one independent variable is called an ordinary differential equation and a differential equation involuting derivatives with respect to more than one independent variables in called partial differential equation. Order of a differential equation is the order of the highest order derivative occurring in the differential equation.

For example - order of differential equation 4  dy  + 3 y d 2 y =  dx   dx 2 ● 0 is 2 Degree of a differential equation is defined if it is a polynomial equation in its derivatives. Degree of the polynomial if defined is the highest power of the highest ordered derivative involved in the differential equation. Degree of a differential equation is a positive integer only. For example - degree of the differential equation d3y dx3 ● 3  d 2 y  + x  2  =  dx  2 0 is 1 A relation between the variables involved in the differential equation which satisfies the given differential equation is called its solution. Solution of differential equation General Solution — A solution which contains as many Particular Solution — A solution which is free from artitary artitary co constants as as th the or order of of th the constants is called is called a particular differential equation is called the solution.

Differential Equations With Applications and Historical Notes (G.F. Simmons) Differential Equations With Applications and Historical Notes (G.F. Renamemaestro 5 4 2 serial port richey. Simmons) DIFFERENTIAL EQUATIONS.

General solution. ● To form a differential equation from a given function, we differentiate the function successively as many times as the number of arbitrary constants in the given function and then eliminate the artitrary constants. ● To from a differential equation representing family of curves given by y 2 = a(b2 − x 2 ), we have to differentiate the relation twice and then eliminate the artitrary constants a and b. ● The order of a differential equation representing a family of curves is equal to the number of arbitrary constant(s) present in the equation representing the family of curves. Types of Differential Equation There are three methods of solving a first order, first degree differential equation depending on its form. These are: (i) Differential equations with variables separable (ii) Homogeneous differential equations (iii) Linear differential equations Let us study each one of them in detail. ● ● ‘Variable Separable Method’ is used to solve such an equation in which variables can be separated completely, i.e., terms containing x should remain with dx and terms containing y should remain with dy.

A function f ( x, y) is said to be a homogeneous function of degree n is (i) f (λx, λy ) = λ n f ( x, y) for some non-zero constant λ. Or  y  or y n h x   y  x   n (ii) f ( x, y ) = x g  Note: A function f ( x, y) is said to be a homogeneous function of degree zero if (i) f (λx, λy ) = f ( x, y ) or  y  or h  x    y  x   (ii) f ( x, y ) = g  ● A homogeneous differential equation of degree zero can be expressed in the form dy dx  y   x = g or  x  = h   dy  y  dx To solve a homogeneous differential equation of the type dy dx = f ( x, y ), we make a substitution y = vx and to solve a homogeneous differential equation of the type dx dy = G( x, y), make a substitution x = vy. ● A differential equation of the form dy dx + Py = Q, where P and Q are either constants or functions of x is know as a first order linear differential equation in y. Solution of such a differential equation is given by y × I.F = ∫ (Q × I.F)dx + C, ∫ where I.F (Integrating Factor) = e Pdx ● A differential equation of the form dx dy + Px = Q, where p and Q are either constants or functions of y is known as first order linear differential in x.

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