The document is a comprehensive guide on ordinary and partial differential equations, focusing on definitions, types, and examples. It includes sections on exact differential equations, their solutions, and various exercises for practice. The content is structured to aid engineers in understanding and applying differential equations in calculus.

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