Guided By:- PROF. DIPESH M. BHOGAYATA Prepared By:- Enrollment No ( Civil Engineering ) First Order & First Degree Ordinary Differential Equation

First Order & First Degree Differential Equation INTRODUCTION :- Differential equations are of fundamental importance in engineering because most of the physical laws and relations appear mathematically are in the form of differential equations.

Definition of differential equation An equation involving differential coefficients is called a differential equation. dy = sinx dx………….(1) ∂z / ∂x + ∂z/ ∂y =0………………….(2) are differential equations.

Classification of differential equations (1) ordinary differential equations (2) partial differential equations

The order and the degree of a differential equation The order of a differential equation is the order of the highest derivative appearing in the differential equation. The degree of a differential equation is the degree of the highest derivative,when the derivatives are free from redicals and fraction. for example, dy = sinx dx ………….(1) equation 1 is first order and first degree equation.

Solution of a differential equation A solution of a differential equation is a relation between the variables, which is a free from derivatives and which setisfies the defferential equation. for example y=A cos x + B sin x …………..(1) is a solution of d²y/dx² + y =0 ………….(2)

Formation of a differential equation Ordinary differential equations are formed by elimination of arbitrary constant. To eliminate two arbitrary constants, we require two more equations besides the given equation,leading us to second order derivatives and hence a differential equation of the second order. In general, elimination of n arbitrary constants will give rise to a differential of the nth order.

solution of differential equation of the first order and first degree  Four types of method are following. Equation In which variables are separable. Homogeneous equation. Linear equation. Exact equation.

Variable separable form If a differential equation of the first order and first degree can be put in the form. F₁(x) dx + F₂(y) dy = 0 then it is called variable separable equation.

Homogeneous equation An equation of the form dy/dx= F₁(x,y)/F₂(x,y) is called a homogeneous differential equation if F₁(x,y) and F₂(x,y) are homogeneous functions of the same degree in x and y.  Method of solution: I. put y = vx there for dy/dx = v+ x dv/dx II.Separate the variables in the new equation formed and solve.

Exact differential equations An equation of the form M(x,y)dx + N(x,y)dy = 0 is called exact if, ∂M/∂y = ∂N/∂x  method of solution is follow: I.Integrate M with respect to x keeping y as constant. II.Integrate only those terms of N with respect to y, which do not contain x. III.Equate the sum of this result is the required solution.

Non exact differential equation An equation of the form M(x,y)dx + N(x,y)dy = 0 is called non exact if, ∂M/∂Y ≠ ∂N/∂X five rules for finding integrating factors.

RULE 2 :-If Mx –Ny ≠ 0 & equation can be written in the form F₁(xy)y dx +F₂(xy)x dy = 0 then 1/Mx-Ny is an I.F RULE 1:- if Mx+Ny ≠ 0 and the equation is homogeneous, then 1/Mx+Ny is an I.F RULE 3 : - for the equation Mdx+Ndy = 0 if 1/N(∂M/∂Y - ∂N/∂X ) is a function of x alone, say f(x) then find I.F RULE 4 :- for the equation Mdx+Ndy = 0 if 1/M(∂N/∂Y - ∂M/∂X ) is a function of y alone, say g(y) then find I.F

Linear differential equation A differential equation is said to be linear if the dependent variable and its derivative occur only in the first degree and are not multiplied together. The form of the L.D.E of the first order is dy/dx + Py = Q