Integration and Differential Equations
Dr Viktor Fedun
v.fedun@sheffield.ac.uk
DifferentiationIntegration 

How to find displacement of an object 

How to find displacement of an object 
Acceleration is the second derivative of the displacement with respect to time, Or the first derivative of velocity with respect to time: 

Inverse procedure: Integration. 


Table of basic integrals 
Definite and indefinite integrals:
Integration Is a Sum from the
by Daniel de Souza Carvalho Integration is a kind of sum. It is easy to realize this by comparing the integration of the function f(x) = 2 with the formula for the area of a rectangle, b x h (base times height). Integration is more general, allowing you to find the area under curves such as a sine wave or a parabola. 
Riemann sum Riemann sum is a method for approximating the total area underneath a curve on a graph, otherwise known as an integral. Riemann Sums: A Simple Illustration from the
Wolfram Demonstrations Project by Phil Ramsden 
Riemann sum Riemann sum is a method for approximating the total area underneath a curve on a graph, otherwise known as an integral. Displacement during time period t_{1}  t_{2} is equal to the definite integral of velocity between limits t_{1} and t_{2} 
Simplest rules of integration 
Find how the velocity v(t) of this object depends upon time.
Acceleration of some physical object is constant:
Find how the position x(t) of this object depends upon time.
Find how the velocity v(t) of this object depends upon time, if v(0)=v_{0}.
Find how the position x(t). of this object depends upon time, if x(0)=x_{0}.
Velocity of some physical object depends upon time as:
Find dependence of displacement S of the same object as a function of time:
Velocity of some physical object depends upon time as:
Find dependence of displacement S of the same object between time moments t_{1} = 1 and t_{2} = 2 as a function of time:
Acceleration of some physical object depends upon time as:
Find dependence of velocity v(t) of the same object as a function of time:
Acceleration of some physical object depends upon time as:
Find the change of velocity v(t) of the same object between time moments t_{1}=5 and t_{2} =10.
Differential equations: First order differential equation is a mathematical relation that relates independent variable, unknown function and the first derivative of unknown function: Above y is an unknown function and t is an independent variable Below we always will assume implicitly that the first derivative can be expressed as a function of the unknown function and the independent variable: A function y=ψ(t) is a solution of the equation above if upon substitution y=ψ(t) into this equation it becomes identity. 
Differential equations: 
Differential equations: Second order differential equation is a mathematical relation that relates independent variable, unknown function, its first derivative and second derivatives In general the order of differential equation is the order of highest derivative of unknown function. Nth order differential equation: 
Differential equations: Let us differentiate this equation: Therefore function e^{at} is a solution of equation: 
Differential equations: Let us differentiate this equation: Therefore function Ce^{at} is a solution of equation: 
Differential equations: Let us solve this equation Therefore function Ce^{at} is a general solution of this equation, i.e. all solutions have this form 
Differential equations: Let us differentiate this equation: Therefore function is a solution of equation 
Differential equations: Let us differentiate this equation: Therefore function is a solution of equation 
Differential equations: Therefore function where C_{1} and C_{2} are arbitrary constants represents general solution of equation 
Geometric Interpretation of the differential equations, Slope Fields Let us consider Cartesian coordinates x and y.
Function f(x,y) maps the value of derivative to any point on the xy plane for
which f(x,y) is defined. Example For plot the associated vectors at the points: (1, 0), (2, 1), (3, 2), (0, 1), (1, 2), where the vectors are pictured having a change in x of 1 (Δx = 1). Solution: 


The vector at the point (1, 0), is to have slope 1,
and it is to have Δx = 1 the vector will be drawn from (1, 0) to (1 + Δx, 0 + Δy), 
Slope Fields from the
Wolfram Demonstrations Project by Charles E. Oelsner


Therefore every point on this plane corresponds to some direction.
This leads to the field of directions. The problem of solving the differential equation can be formulated as follows: Any particular integral curve represents a particular solution of differential equation. Mathematical relation that describes the whole family of integral curves for any given equation corresponds to the general solution of this equation 
Direction Fields for Differential Equations 
Direction Fields for Differential Equations
from the
Wolfram Demonstrations Project by Stephen Wilkerson

First order linear differential equations 
First order linear differential equation with constant coefficients is a linear equation with respect of unknown function and its derivative: Where coefficients A≠0 and B are constants and do not depend upon x. In general case coefficient C does depend x. It is customary in mathematics to write the equation above as: 
First order linear differential equations 
If Q(x)=0 The equation is called the first order linear homogeneous equation. 
Example. Solution of the first order linear homogeneous equations 
A solution of a differential equation with its constants undetermined
is called Linear Differential Equations (Essential calculus by James Stewart) click here ! 
Solution of the first order linear nonhomogeneous equations 
Let us try to find solution of in the same form as for homogeneous equation, but with parameter C that depends upon t. 
Solution of the first order linear nonhomogeneous equations 


Nonhomogeneous Linear Equations (Essential calculus by James Stewart) click here ! 
Solution of the first order linear nonhomogeneous equations 


Solution of the first order linear nonhomogeneous equations 


Solution of the first order linear nonhomogeneous equations 



Solution of the first order linear nonhomogeneous equations 




Response of a linear system to a periodic input 



Solution of the first order linear nonhomogeneous equations 
If we know one particular solution of nonhomogeneous linear equation we can reduce this equation to a homogeneous one. Let us assume that Y is some known solution of nonhomogeneous equation. Let us introduce new unknown function z: y = Y + z Let us substitute this new variable into nonhomogeneous equation: 
Solution of the first order linear nonhomogeneous equations 

Summary: How to find the solution of first order, linear homogeneous, differential equations with constant coefficients? 

How to find the solution of second order, linear, homogeneous differential equation with constant coefficients? 

2nd order Linear Differential Equations with constant coefficients 

2nd order Linear Differential Equations with constant coefficients 

2nd order Linear Differential Equations with constant coefficients 

2nd order Linear Differential Equations with constant coefficients 

2nd order Linear Differential Equations with constant coefficients 

Summary: How to find the solution of second order, linear, homogeneous differential equations with constant coefficients? 

Second order linear differential equations 

Solution of second order, linear, nonhomogeneous equations 
Second order linear differential equations (Essential calculus by James Stewart) click here ! 
How to find a particular solution of a second order nonhomogeneous differential
equation with constant coefficients? 
Let us consider some second order linear differential nonhomogeneous equation: Let us assume that we know the general solution of corresponding homogeneous equation: How can we find a particular solution to this nonhomogeneous equation? 
How to find a particular solution of a second order nonhomogeneous differential
equation with constant coefficients? 
Let us try to find a particular solution of this nonhomogeneous equation in the same form as the general solution of the corresponding homogeneous equation but with constants C1(t) and C2(t) that depend upon t. 
“Variations of constants method”: 



“Variations of constants method”: 



Let us verify this result: 

“Variations of constants method”: 




Let us verify this result: 

References 
Essential calculus by James Stewart click here ! 