Laplace Transform Example

Laplace Transform ExampleAbstract This paper describes the Laplace transform utilize in closure the derivative equivalence and the comparison with the other usual regularity actings of solving the derived involvement gear par. The method of Laplace transform has the advantage of directly big(a) the resolvent of differential equating with aband stard barrier value without the necessary of beginning-class honours degree finding the cosmopolitan effect and then evaluating from it the arbitrary constants. except the ready formulas of the Laplace reduce the problem of solving differential comparisons to mere algebraic manipulation.IntroductionDifferential comparison is an equation which involves differential coefficients or differentials. It may be defined in a more refined sort as an equation that defines a Relationship between a break and angiotensin-converting enzyme or more derivatives of that run away. Let y be nearly affair of the independent variable t. Then following be around differential equations relating y to wholeness or more of its derivatives.The equation states that the first derivative of the function y equals the product of and the function y itself. An additional, unverbalised statement in this differential equation is that the stated relationship holds all for all t for which both(prenominal) the function and its first derivative be defined. Some other differential equations Differential equations arise from more problems in oscillations of mechanical and electrical systems, bending of beams conduction of heat, velocity of chemical reactions etc., and as such play a very important role in all modern scientific and engineering studies. There argon many instructions of solving the differential equation and the well-nigh effective way is to lend atomic number 53self the Laplace equation because it provides the easy path to solve the differential equation without involving any long process of finding out the co mplementary function and particular integral.Solution of differential equationA dissolving agent of a differential equation is a relation between the variables which satisfy the disposed differential equation. A first order homogeneous differential equation involves sole(prenominal) the first derivative of a function and the function itself, with constants only as multipliers. The equation is of the formand can be solved by the substitutio The solvent which fits a specific fleshly situation is obtained by replace the event into the equation and evaluating the various constants by forcing the solution to fit the physical boundary powers of the problem at hand. Substituting givesThe general solution to a differential equation essential satisfy both the homogeneous and non-homogeneous equations. It is the record of the homogeneous solution that the equation gives a zero value. If you find a particular solution to the non-homogeneous equation, you can add the homogeneous solut ion to that solution and it lead still be a solution since its net resolution will be to add zero. This does not mean that the homogeneous solution adds no meaning to the picture the homogeneous part of the solution for a physical situation helps in the at a lower placestanding of the physical system. A solution can be formed as the sum of the homogeneous and non-homogeneous solutions, and it will have a number of arbitrary (undetermined) constants. Such a solution is called the general solution to the differential equation. For performance to a physical problem, the constants must be determined by forcing the solution to fit physical boundary conditions. Once a general solution is formed and then force to fit the physical boundary conditions, one can be assured that it is the unique solution to the problem, as gauranteed by the uniqueness theorem.Uniqueness theoremFor the differential equations applicable to physical problems, it is often possible to start with a general form and force that form to fit the physical boundary conditions of the problem. This broad of approach is made possible by the fact that there is one and only one solution to the differential equation, i.e., the solution is unique.Stated in basis of a first order differential equation, if the problem meets the condition such that f(x,y) and the derivative of y is continuous in a given rectangle of (x,y) values, then there is one and only one solution to the equation which will meet the boundary conditions.Laplace in solving differential equationThe Laplace transform method of solving differential equations yields particular solutions without the necessity of first finding the general solution and then evaluating the arbitrary constants. This method is in general shorter than the above mentioned methods and is specially used for solving the linear differential equation with constant coefficients.Working procedure Take the Laplace transform of both sides of the differential equation exploitation the formulas of Laplace and the given initial conditions. Transpose the terms with minus sign to right. Divide by the coefficient of y, charterting y as a known function of s. Resolve this function of s into partial derivative fractions and take the inverse transform of both sides.This gives y as a function of t which is the in demand(p) solution satisfying the given conditions. work the algebraic equation in the mapped spaceBack transformation of the solution into the original space. Figure 1 Schema for solving differential equations using the Laplace transformationSome of the instances which demonstrate the use of the Laplace in solving the differential equation are as followsExample no.1 Consider the differential equation with the initial conditions . Proceeding using the steps given above one has Step 1 Step 2 Step 3 The building complex function must be decomposed into partial fractions in order to use the tables of correspondences. This gives By using the for mulas of the inverse laplace transform we can convert these relative frequency domains back in the time domain and hence get the desired result as , Another example of the laplace involving trigonometric function is We expect to solvewith initial conditions f(0) = 0 and f (0)=0.We note thatand we getSo this is equivalent toWe deduceSo we apply the Laplace inverse transform and getPeriodic functionsIn mathematics, a mensesic function is a function that repeats its values in regular intervals or levels. The some important examples are the trigonometric functions, which repeat over intervals of length 2. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit bimestriality.A function f is said to be periodic iffor all values of x. The constant P is called the period, and is required to be nonzero. A function with period P will repeat on intervals of length P, and these intervals are sometimes also referred to as periods.For example, the sine function is periodic with period 2, sincefor all values of x. This function repeats on intervals of length 2 (see the represent to the right).Geometrically, a periodic function can be defined as a function whose graph exhibits translational symmetry. Specifically, a function f is periodic with period P if the graph of f is invariant under translation in the x-direction by a distance of P. This definition of periodic can be extended to other geometric shapes and patterns, such as periodic tessellations of the plane.A function that is not periodic is called aperiodic.Laplace transform of periodic functionsIf function f(t) is periodic with period p 0, so that f(t + p) = f(t), and f1(t) is one period (i.e. one cycle) of the function, then the Laplace of this periodic function is given by The basic model of the formula is the Laplace Transform of the periodic function f(t) with period p, equals the Laplace Transform of one cycle of the function, divided by (1 e-sp) .Laplace transform of some of the common functions want the graph given below is given by Fig no3continous graphical functionFrom the graph, we see that the first period is given by and that the period p = 2.NowSoHence, the Laplace transform of the periodic function, f(t) is given byOther continuous wave forms and there Laplace transforms areThis wave is an example of the full wave rectification which is obtained by the rectifier used in the electronic instruments.Here,and the period, p = .So the Laplace Transform of the periodic function is given byConclusionThe knowledge of Laplace transform has in recent years function an essential part of mathematical background required of engineers and scientists. This is because the transform method an easy and effective means for the solution of many problems arising in engineering. The method of laplace transformation is proving to be the most effective and easy way of solving differential equations and hence it is replacing other methods of solution of the differential equation. The most frequent function encompassed in electronics engineering is continuous function and most of the functions are in the time domain and we need to convert them in the frequency domain, this operation is performed excellently by the Laplace transform and hence its application is further enlarged using it in the solution of the continuous functions.