G.H.PATEL COLLEGE OF ENGINEERING AND TECHNOLOGY

Chemical Department

 Topic Name: First Order ODE with Application

 Subject: Advanced Engineering Mathematics(2130002)

 Name:

Shivam Patel (150110105031)Vatsal Patel (150110105032)Vyom Patodiya (150110105033)Krishna Peshivadiya (150110105034)Hardik Pipaliya (150110105035)Bhavin Poshiya (150110105036).

 Faculty Name: Prof. Mukesh T. Joshi

Content:IntroductionSome useful TermsType of First Order ODEsBernoulli EquationOrthogonal Trajectory

First Order Ordinary Linear Differential Equations

Ordinary Differential equations does not include partial derivatives.

A linear first order equation is an equation that can be expressed in the form

Where p and q are functions of x

Some useful Terms:

First order differential equation with y as the dependent variable and x as the independent variable would be:

Second order differential equation will have the form:

dy f x,ydx

),,(2

2

dxdyyxf

dxyd

The order of a differential equation is the order of the highest derivative occurring in the equation.

The degree of a differential equation is the power of the highest ordered derivative occurring in the equation, provided all derivatives are made free from radicals and fractions.

• Uniqueness: Does a differential equation have more than one solution? If yes, how can we find a solution which satisfies particular conditions?

If no initial conditions are given, we call the description of all solutions to the differential equation the general solution. This type of solution will contain arbitrary constants whose number is equal to the order of differential equation.

A solution obtained from the general solution by giving particular values to arbitrary constants is known as a particular solution.

A solution which cannot be obtained from general solution is called a singular solution.

Types Of First 0rder ODE:

1. Variable Separable2. Homogeneous Differential Equation3. Exact Differential Equation4. Linear Differential Equation

The first-order differential equation

,dy f x ydx

is called separable provided that f(x,y) can be written as the product of a function of x and a function of y.

(1)

1. Variable Separation:-

Suppose we can write the above equation as

)()( yhxgdxdy

We then say we have “ separated ” the variables. By taking h(y) to the LHS, the equation becomes

1 ( )( )

dy g x dxh y

Integrating, we get the solution as

1 ( )( )

dy g x dx ch y

where c is an arbitrary constant.

Example 1. Consider the D.E. ydxdy

Sol. Separating the variables, we get

dxdyy

1

Integrating we get the solution as

kxy ||ln

or ccey x , an arbitrary constant.

2. Homogeneous Diff. equations

Definition : A function f(x, y) is said to be homogeneous of degree n in x, y if

),(),( yxfttytxf n for all t, x, y

Examples 22 2),( yxyxyxf is homogeneous of degree

)sin(),(x

xyxyyxf

is homogeneous of degree

2.

0.

A first order D.E. 0),(),( dyyxNdxyxM

is called homogeneous if ),(,),( yxNyxM

are homogeneous functions of x and y of the same degree.

This D.E. can be written in the form

),( yxfdxdy

The substitution y = vx converts the given equation into “variables separable” form and hence can be solved. (Note that v is also a new variable)

Working Rule to solve homogeneous ODE:1. Put the given equation in the form

.Let .3 zxy

)1(0),(),( dyyxNdxyxM

2. Check M and N are Homogeneous function of the same degree.

5. Put this value of dy/dx into (1) and solve the equation for z by separating the variables.

6. Replace z by y/x and simplify.

dxdzxz

dxdy

4. Differentiate y = z x to get

Example:- Solve the Homogeneous ODE

)sin(x

xyxyy

Let y = z x. Hence we get

zzdxdzxz sin or z

dxdzx sin

Separating the variables, we getx

dxdzz

sin

1

Integrating we get

where

cosec z – cot z = c x

and c an arbitrary constant.yzx

3. EXACT DIFFERENTIAL EQUATIONS

A first order D.E. 0),(),( dyyxNdxyxMis called an exact D.E. if

M Ny x

The solution given by:

cNdyMdx

(Terms free from ‘x’)

Solution is:

cyxy

cdyy

ydx

ln2

2

cNdyMdx

Example 1 The D.E. 0x dx y dy

Sol.:- It is exact as it is d (x2+ y2) = 0

Hence the solution is: x2+ y2 = c

Example 2 The D.E. 2

1 sin( ) sin( ) 0x x xdx dyy y y y

Sol:- It is exact (cos( )) 0xdy

Hence the solution is: cos( )x cy

Integrating FactorsDefinition: If on multiplying by I.F.(x, y) or , the D.E.

0M dx N dy

becomes an exact D.E. , we say that I.F.(x, y) or is an Integrating Factor of the above D.E.

2 2

1 1 1, ,xy x y

are all integrating factors of

the non-exact D.E. 0y dx x dy

( )

M Ny x g x

N

( )g x dxe

is an integrating factor of the given DE

0M dx N dy

Rule 1: I.F is a function of ‘x’ alone.

Rule 2: I.F is a function of ‘y’ alone.

If , a function of y alone,

then( )h y dy

e is an integrating factor of the given DE.

xN

yM

M yh

is an integrating factor of the given DE

0M dx N dy

Rule 3: Given DE is homogeneous.

NyMx1

Rule 4: Equation is of the form of

021 xdyxyfydxxyf

Then,

NyMx

1

Example 1 Find an I.F. for the following DE and hence solve it.

2( 3 ) 2 0x y dx xy dy Sol.:-Here

6 2M Ny yy x

2( 3 ); 2M x y N xy

Hence the given DE is not exact.

M Ny x

N

Now 6 2

2y y

xy

2 ( ),g xx

a function of x alone. Hence

( )g x dxe

22dx

xe x is an integrating factor of the given DE

Multiplying by x2, the given DE becomes

3 2 2 3( 3 ) 2 0x x y dx x y dy

which is of the form 0M dx N dy

Note that now 3 2 2 3( 3 ); 2M x x y N x y

Integrating, we easily see that the solution is

43 2 ,

4x x y c c an arbitrary constant.

4. Linear Diff. Equations

A linear first order equation is an equation that can be expressed in the form

1 0( ) ( ) ( ), (1)dya x a x y b xdx

where a1(x), a0(x), and b(x) depend only on the independent variable x, not on y.

We assume that the function a1(x), a0(x), and b(x) are continuous on an interval and that a1(x) 0on that interval. Then, on dividing by a1(x), we can rewrite equation (1) in the standard form

( ) ( ) (2)dy P x y Q xdx

where P(x), Q(x) are continuous functions on the interval.

Rules to solve a Linear D.E.:-

1. Write the equation in the standard form

( ) ( )dy P x y Q xdx

2. Calculate the IF (x) by the formula

( ) exp ( )x P x dx

3. Multiply the equation by (x).

4. Integrate the last equation.

Example 1. Solve

1cossincos

xxxy

dxdyxx

Solution :- Dividing by x cos x, throughout, we get

xxy

xx

dxdy sec1tan

1tan( )( )x dxP x dx xx e e

log sec log loglog sec( ) sece e ex x xxx e e e x x

secsec secd xdx x

y x x x x

2sec sec tany x x dx C x C

Multiply by secx x so

Integrate both side we get

Bernoulli Equation

In mathematics, an ordinary differential equation of the form:

y'+P(x)y=Q(x)y^{n}

is called a Bernoulli differential equation where { n} is any real number and  n≠ 0 or  n ≠1. It is named after Jacob Bernoulli who discussed it in 1695.

Bernoulli equations are special because they are nonlinear differential equations with known exact solutions. 

before. told weassolution its andequation linear is this

)()1()()1(

)()()1(

1

)1( then Put b)

)()(1yover Equation Bernoulli Divide a)

Equation Bernoulli solve To

)1(

)1(

n

xQnzxpndxdz

xQxpdxdz

n

dxdyyn

dxdzyz

xQ yx Pdxdy

y

n

nn

note :- if n = 0 the Bernoulli Equation will be linear equation.

if n = 1 Bernoulli Equation will be separable equation

the general solution will be

2

22

1

2

lnexpln2expexpexp

linear isequation thissin62

22

.sin3

solution

sin3 )

x

xxdxpdx

x xz

dxdz

dxdyy

dxdzen and th yput z

y xxy –

dxdy

xxy –

dxdyy Ex

x

c x x x x y x

cxdx x y x

)cossincos(6

sin622

22

Orthogonal TrajectoriesOrthogonal trajectories are a

family of curves in the plane that intersect a given family of curves at right angles.

Suppose we are given the family of circles centered about the origin:x2 + y2 = c The orthogonal trajectories of this family are the family of curves/lines such that intersect the circle at right angles.

ExampleFind the orthogonal trajectories of the curve x2

+ y2 = c2

Solution:-Differentiating w.r.t. variable x, we get

Writing it in explicit form, we get

The derived equation is a linear equation.If we use the method of solving linear

equations, then we derive the following results for I.F. :

Thus, we get the general solution asy = mx

General Applications of ODEODEs are used in various fields such as:

Radioactivity and carbon datingEquations of series RL circuitsEconomics Bernoulli EquationPopulation DynamicsNewton’s Law of Cooling

•Cooling/Warming lawWe have seen in Newton’sempirical law of cooling of

an object in given by the linear first-order differential equation

)mTα(TdtdT

This is a separable differential equation. We have

αdt)T(T

dT

m

or ln|T-Tm |

=t+c1

or T(t) = Tm+c2et

In Series Circuits

i(t), is the solution of the differential equation.

)(tERidtdi

dtdqi

)t(EqC1

dtdqR

Since

It can be written as