Integration and Differential Equations

Dr Viktor Fedun

v.fedun@sheffield.ac.uk

Differentiation-Integration



How to find displacement of an object
if we know its acceleration as a function of time?



How to find displacement of an object
if we know its acceleration as a function of time?

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.
Velocity is an integral of acceleration over time.
Displacement is an integral of velocity over time.







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 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 t1 - t2 is equal to the definite integral of velocity between limits t1 and t2



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)=v0.





Find how the position x(t). of this object depends upon time, if x(0)=x0.











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 t1 = 1 and t2 = 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 t1=5 and t2 =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. N-th order differential equation:





Differential equations:





Let us differentiate this equation:





Therefore function eat is a solution of equation:





Differential equations:





Let us differentiate this equation:





Therefore function Ceat is a solution of equation:





Differential equations:





Let us solve this equation





Therefore function Ceat 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 C1 and C2 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 x-y plane for which f(x,y) is defined.
The curve y=ψ(x) is called an integral curve of the differential equation if y=ψ(x) is a solution of this equation. The derivative of y with respect to x determines the direction of the tangent line to this curve. It is equal to tan(α) where α is an angle between the tangent line and the x-axis.



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:



(x,y)

Δx

(x+Δx, y+Δy)
(1,0) 1 1 1 (2, 1)
(2,1) 1 1 1 (3, 2)
(3,2) 1 1 1 (4, 3)
(0,1) -1 1 -1 (1, 0)
(-1,2) -3 1 -3 (0, -1)


The vector at the point (1, 0), is to have slope 1, and it is to have Δx = 1
Since



the vector will be drawn from (1, 0) to (1 + Δx, 0 + Δy),
so it will have terminal point at (1 + 1, 0 + 1) = (2, 1).

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:
Find a curve such that at any point on this curve the direction of the tangent line corresponds to the field of direction for this equation.



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



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.
If Q(x)≠0 the equation is called the first order linear non-homogeneous equation.



Example. Solution of the first order linear homogeneous equations







A solution of a differential equation with its constants undetermined is called
a general solution.





Linear Differential Equations (Essential calculus by James Stewart) click here !



Solution of the first order linear non-homogeneous equations
'Variations of constants method'





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 non-homogeneous equations
'Variations of constants method'











Non-homogeneous Linear Equations (Essential calculus by James Stewart) click here !



Solution of the first order linear non-homogeneous equations
'Variations of constants method'







Solution of the first order linear non-homogeneous equations
'Variations of constants method'







Solution of the first order linear non-homogeneous equations
'Variations of constants method'









Solution of the first order linear non-homogeneous equations
'Variations of constants method'













Response of a linear system to a periodic input









Solution of the first order linear non-homogeneous equations



If we know one particular solution of non-homogeneous linear equation we can reduce this equation to a homogeneous one. Let us assume that Y is some known solution of non-homogeneous equation. Let us introduce new unknown function z: y = Y + z



Let us substitute this new variable into non-homogeneous equation:





Solution of the first order linear non-homogeneous 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
Examples:
Homogeneous Equation two distinct real roots



2nd order Linear Differential Equations with constant coefficients
Examples:
Homogeneous Equation two coincident real roots



2nd order Linear Differential Equations with constant coefficients
Examples:
Homogeneous Equation two imaginary roots



2nd order Linear Differential Equations with constant coefficients
Examples:
Homogeneous Equation two complex roots



2nd order Linear Differential Equations with constant coefficients
Homogeneous Equation two complex roots general case



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, non-homogeneous equations





Second order linear differential equations (Essential calculus by James Stewart) click here !



How to find a particular solution of a second order non-homogeneous differential equation with constant coefficients?
“Variations of constants method”

Let us consider some second order linear differential non-homogeneous equation:





Let us assume that we know the general solution of corresponding homogeneous equation:







How can we find a particular solution to this non-homogeneous equation?



How to find a particular solution of a second order non-homogeneous differential equation with constant coefficients?
“Variations of constants method”

Let us try to find a particular solution of this non-homogeneous 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”:
Example 1









“Variations of constants method”:
Example 1







Let us verify this result:





“Variations of constants method”:
Example 2









Let us verify this result:





References

Essential calculus by James Stewart click here !