Exact Differential Equations

Prasad Enagandula
Prasad EnagandulaAsst.Professor at Vignana bharathi institute of technology à Vignana bharathi institute of technology
Ordinary Differential Equations
First Order and First Degree
Dr.E.Prasad
CALCULUS FOR ENGINEERS DIFFERENTIAL EQUATIONS
Dr.E.Prasad, Assoc Professor
CONTENTS
CALCULUS FOR ENGINEERS DIFFERENTIAL EQUATIONS
Dr.E.Prasad, Assoc Professor
1.1.Introduction
1.2.Exact Differential Equations
1.3.Non Exact Differential Equations
1.4.Linear Differential Equations
1.5.Bernoulli’s Differential Equations
1.6.Applications of First order Differential Equations
1.1.Introduction
CALCULUS FOR ENGINEERS DIFFERENTIAL EQUATIONS
Dr.E.Prasad, Assoc Professor
Basics
A differential equation is an equation which contains the derivatives of one variable (i.e.,
dependent variable) with respect to the other variable (i.e., independent variable)
Examples :
1. (dy/dx) = sin x
2. (d2y/dx2) + k2y = 0
3. (∂2z/∂s2) + (∂2z/∂t2) = 0
4. (d3y/dx3) + x(dy/dx) - 4xy = 0
5. (rdr/dθ) + cosθ = 5
Ordinary Differential Equation
An ordinary differential equation involves function and its derivatives. It contains only one
independent variable and one or more of its derivatives with respect to the variable.
Examples :
1. (dy/dx) + (d2y/dx2) +y2 + 2x = 0 is ordinary differential equation
2. (d2y/dx2) + e2xy = 0
Differential Equation
CALCULUS FOR ENGINEERS 1.1.Introduction
Dr.E.Prasad, Assoc Professor
Basics
A Partial Differential Equation commonly known as PDE is a differential equation containing
partial derivatives of the dependent variable (one or more) with more than one independent
variable
Examples:
1.
The Order of a differential equation is the order of the highest derivative
(also known as differential coefficient) present in the equation.
Example (i):(d3y/dx3) + x(dy/dx) - 4xy = 0
In this equation, the order of the highest derivative is 3 hence, this is a third order
differential equation.
Example (ii) This equation represents a second order differential equation.
(dy/dx) + (d2y/dx2) +y2 + 2x = 0
Partial Differential Equation
Order of the Differential Equation
CALCULUS FOR ENGINEERS 1.1.Introduction
Dr.E.Prasad, Assoc Professor
Basics
The degree of the differential equation is represented by the power of the highest order
derivative in the given differential equation and free from reciprocals
The differential equation must be a polynomial equation in derivatives for the degree to be
defined.
Examples:
1.7(d4y/dx4)3 + 5(d2y/dx2)4+ 9(dy/dx)8 + 11y = 0. This differential equation is of fourth order
and a degree of three.
2.(dy/dx)2 + (dy/dx) - Cos3x = 0. This differential equation is first order and of second degree.
3.(d2y/dx2) 1/2 + x(dy/dx)3 = 0. This differential equation is of second order and the first
degree.(how..)
Degree of the Differential Equation
CALCULUS FOR ENGINEERS 1.1.Introduction
Dr.E.Prasad, Assoc Professor
1.2.Exact Differential Equations
CALCULUS FOR ENGINEERS DIFFERENTIAL EQUATIONS
Dr.E.Prasad, Assoc Professor
Definition:
A differential equation of the type M(x,y)dx + N(x,y) dy = 0 can be said as an exact
differential equation if there exist a function of two variables f(x,y)having continuous
partial derivatives such that
fx(x,y) = M(x,y) and
fy(x,y) = N(x,y)
Thus ,the general solution of the equation is f(x,y) = C , where C is any arbitrary
constant .
Exact Differential Equation
Test for exactness of the differential equation
The necessary and sufficient conditions for O.D.E of the form M(x,y)dx + N(x,y) dy = 0
to be exact is
CALCULUS FOR ENGINEERS 1.2.Exact Differential Equations
Dr.E.Prasad, Assoc Professor
In order to obtain the solution of an Exact differential equation, we have
to proceed as follows:
1. Integrate M with respect to x, keeping y as constant.
2. Integrate with respect to y only those terms of N which do not contain
x.
3. Add the two expressions obtained in (1) and (2) above and equate the
result to an arbitrary constant.
In other words, the solution of an exact differential equation is
General Solution-Exact Differential Equation
Exact D.E
CALCULUS FOR ENGINEERS 1.2.Exact Differential Equations
Dr.E.Prasad, Assoc Professor
1.
Problems-Exact Differential Equation
CALCULUS FOR ENGINEERS 1.2.Exact Differential Equations
Dr.E.Prasad, Assoc Professor
CALCULUS FOR ENGINEERS 1.2.Exact Differential Equations
Dr.E.Prasad, Assoc Professor
2.
CALCULUS FOR ENGINEERS 1.2.Exact Differential Equations
Dr.E.Prasad, Assoc Professor
1.Solve 1 + 𝑒
𝑥
𝑦 𝑑𝑥 + 𝑒
𝑥
𝑦 1 −
𝑥
𝑦
𝑑𝑦 = 0
2.Solve 𝑥2
− 4𝑥𝑦 − 2𝑦2
𝑑𝑥 + 𝑦2
− 4𝑥𝑦 − 2𝑥2
𝑑𝑦 = 0
3.Solve 2𝑥𝑦 𝑑𝑦 − 𝑥2 − 𝑦2 + 1 𝑑𝑥 = 0
4.Solve(3x2
y3
+ 5x4
y2
) dx + (3 y2
x3
+ 2x5
y) dy = 0
5.Solve 𝑦𝑐𝑜𝑠𝑥 + 𝑠𝑖𝑛𝑦 + 𝑦 𝑑𝑥 + 𝑠𝑖𝑛𝑥 + 𝑥𝑐𝑜𝑠𝑦 + 𝑥 𝑑𝑦 = 0
6.Solve 𝑥𝑒𝑥𝑦
+ 2𝑦
𝑑𝑦
𝑑𝑥
+ 𝑦𝑒𝑥𝑦
= 0
Exercise-Exact Differential Equation
CALCULUS FOR ENGINEERS 1.2.Exact Differential Equations
Dr.E.Prasad, Assoc Professor
1. x+𝑦𝑒
𝑥
𝑦 = 𝑐
5. ysinx + siny + y = c
6. 𝑦2
+ 𝑒𝑥𝑦
= 𝑐
4. x3
y2
(y + x2
) = c
Exercise-Solutions
2. x3
+ y3
-6xy(y +x) = c
3. 𝑥3
− 3xy2
+3x = c
CALCULUS FOR ENGINEERS 1.2.Exact Differential Equations
Dr.E.Prasad, Assoc Professor
1 sur 14

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Exact Differential Equations

  • 1. Ordinary Differential Equations First Order and First Degree Dr.E.Prasad CALCULUS FOR ENGINEERS DIFFERENTIAL EQUATIONS Dr.E.Prasad, Assoc Professor
  • 2. CONTENTS CALCULUS FOR ENGINEERS DIFFERENTIAL EQUATIONS Dr.E.Prasad, Assoc Professor 1.1.Introduction 1.2.Exact Differential Equations 1.3.Non Exact Differential Equations 1.4.Linear Differential Equations 1.5.Bernoulli’s Differential Equations 1.6.Applications of First order Differential Equations
  • 3. 1.1.Introduction CALCULUS FOR ENGINEERS DIFFERENTIAL EQUATIONS Dr.E.Prasad, Assoc Professor
  • 4. Basics A differential equation is an equation which contains the derivatives of one variable (i.e., dependent variable) with respect to the other variable (i.e., independent variable) Examples : 1. (dy/dx) = sin x 2. (d2y/dx2) + k2y = 0 3. (∂2z/∂s2) + (∂2z/∂t2) = 0 4. (d3y/dx3) + x(dy/dx) - 4xy = 0 5. (rdr/dθ) + cosθ = 5 Ordinary Differential Equation An ordinary differential equation involves function and its derivatives. It contains only one independent variable and one or more of its derivatives with respect to the variable. Examples : 1. (dy/dx) + (d2y/dx2) +y2 + 2x = 0 is ordinary differential equation 2. (d2y/dx2) + e2xy = 0 Differential Equation CALCULUS FOR ENGINEERS 1.1.Introduction Dr.E.Prasad, Assoc Professor
  • 5. Basics A Partial Differential Equation commonly known as PDE is a differential equation containing partial derivatives of the dependent variable (one or more) with more than one independent variable Examples: 1. The Order of a differential equation is the order of the highest derivative (also known as differential coefficient) present in the equation. Example (i):(d3y/dx3) + x(dy/dx) - 4xy = 0 In this equation, the order of the highest derivative is 3 hence, this is a third order differential equation. Example (ii) This equation represents a second order differential equation. (dy/dx) + (d2y/dx2) +y2 + 2x = 0 Partial Differential Equation Order of the Differential Equation CALCULUS FOR ENGINEERS 1.1.Introduction Dr.E.Prasad, Assoc Professor
  • 6. Basics The degree of the differential equation is represented by the power of the highest order derivative in the given differential equation and free from reciprocals The differential equation must be a polynomial equation in derivatives for the degree to be defined. Examples: 1.7(d4y/dx4)3 + 5(d2y/dx2)4+ 9(dy/dx)8 + 11y = 0. This differential equation is of fourth order and a degree of three. 2.(dy/dx)2 + (dy/dx) - Cos3x = 0. This differential equation is first order and of second degree. 3.(d2y/dx2) 1/2 + x(dy/dx)3 = 0. This differential equation is of second order and the first degree.(how..) Degree of the Differential Equation CALCULUS FOR ENGINEERS 1.1.Introduction Dr.E.Prasad, Assoc Professor
  • 7. 1.2.Exact Differential Equations CALCULUS FOR ENGINEERS DIFFERENTIAL EQUATIONS Dr.E.Prasad, Assoc Professor
  • 8. Definition: A differential equation of the type M(x,y)dx + N(x,y) dy = 0 can be said as an exact differential equation if there exist a function of two variables f(x,y)having continuous partial derivatives such that fx(x,y) = M(x,y) and fy(x,y) = N(x,y) Thus ,the general solution of the equation is f(x,y) = C , where C is any arbitrary constant . Exact Differential Equation Test for exactness of the differential equation The necessary and sufficient conditions for O.D.E of the form M(x,y)dx + N(x,y) dy = 0 to be exact is CALCULUS FOR ENGINEERS 1.2.Exact Differential Equations Dr.E.Prasad, Assoc Professor
  • 9. In order to obtain the solution of an Exact differential equation, we have to proceed as follows: 1. Integrate M with respect to x, keeping y as constant. 2. Integrate with respect to y only those terms of N which do not contain x. 3. Add the two expressions obtained in (1) and (2) above and equate the result to an arbitrary constant. In other words, the solution of an exact differential equation is General Solution-Exact Differential Equation Exact D.E CALCULUS FOR ENGINEERS 1.2.Exact Differential Equations Dr.E.Prasad, Assoc Professor
  • 10. 1. Problems-Exact Differential Equation CALCULUS FOR ENGINEERS 1.2.Exact Differential Equations Dr.E.Prasad, Assoc Professor
  • 11. CALCULUS FOR ENGINEERS 1.2.Exact Differential Equations Dr.E.Prasad, Assoc Professor
  • 12. 2. CALCULUS FOR ENGINEERS 1.2.Exact Differential Equations Dr.E.Prasad, Assoc Professor
  • 13. 1.Solve 1 + 𝑒 𝑥 𝑦 𝑑𝑥 + 𝑒 𝑥 𝑦 1 − 𝑥 𝑦 𝑑𝑦 = 0 2.Solve 𝑥2 − 4𝑥𝑦 − 2𝑦2 𝑑𝑥 + 𝑦2 − 4𝑥𝑦 − 2𝑥2 𝑑𝑦 = 0 3.Solve 2𝑥𝑦 𝑑𝑦 − 𝑥2 − 𝑦2 + 1 𝑑𝑥 = 0 4.Solve(3x2 y3 + 5x4 y2 ) dx + (3 y2 x3 + 2x5 y) dy = 0 5.Solve 𝑦𝑐𝑜𝑠𝑥 + 𝑠𝑖𝑛𝑦 + 𝑦 𝑑𝑥 + 𝑠𝑖𝑛𝑥 + 𝑥𝑐𝑜𝑠𝑦 + 𝑥 𝑑𝑦 = 0 6.Solve 𝑥𝑒𝑥𝑦 + 2𝑦 𝑑𝑦 𝑑𝑥 + 𝑦𝑒𝑥𝑦 = 0 Exercise-Exact Differential Equation CALCULUS FOR ENGINEERS 1.2.Exact Differential Equations Dr.E.Prasad, Assoc Professor
  • 14. 1. x+𝑦𝑒 𝑥 𝑦 = 𝑐 5. ysinx + siny + y = c 6. 𝑦2 + 𝑒𝑥𝑦 = 𝑐 4. x3 y2 (y + x2 ) = c Exercise-Solutions 2. x3 + y3 -6xy(y +x) = c 3. 𝑥3 − 3xy2 +3x = c CALCULUS FOR ENGINEERS 1.2.Exact Differential Equations Dr.E.Prasad, Assoc Professor