Laplace transform applied to differential equations
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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the
Laplace transform In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform In mathematics, an integral transform maps a function from its original function space into another function space via integra ...
is a powerful
integral transform In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in ...
used to switch a function from the
time domain Time domain refers to the analysis of mathematical functions, physical signals or time series of economic or environmental data, with respect to time. In the time domain, the signal or function's value is known for all real numbers, for the cas ...
to the
s-domain In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the ''time domain'') to a function of a complex variable s (in the com ...
. The Laplace transform can be used in some cases to solve
linear differential equation In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b( ...
s with given
initial conditions In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted ''t'' = 0). For ...
. First consider the following property of the Laplace transform: :\mathcal\=s\mathcal\-f(0) :\mathcal\=s^2\mathcal\-sf(0)-f'(0) One can prove by
induction Induction, Inducible or Inductive may refer to: Biology and medicine * Labor induction (birth/pregnancy) * Induction chemotherapy, in medicine * Induced stem cells, stem cells derived from somatic, reproductive, pluripotent or other cell t ...
that :\mathcal\=s^n\mathcal\-\sum_^s^f^(0) Now we consider the following differential equation: :\sum_^a_if^(t)=\phi(t) with given initial conditions :f^(0)=c_i Using the
linearity Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
of the Laplace transform it is equivalent to rewrite the equation as :\sum_^a_i\mathcal\=\mathcal\ obtaining :\mathcal\\sum_^a_is^i-\sum_^\sum_^a_is^f^(0)=\mathcal\ Solving the equation for \mathcal\ and substituting f^(0) with c_i one obtains :\mathcal\=\frac The solution for ''f''(''t'') is obtained by applying the
inverse Laplace transform In mathematics, the inverse Laplace transform of a function ''F''(''s'') is the piecewise-continuous and exponentially-restricted real function ''f''(''t'') which has the property: :\mathcal\(s) = \mathcal\(s) = F(s), where \mathcal denotes the ...
to \mathcal\. Note that if the initial conditions are all zero, i.e. :f^(0)=c_i=0\quad\forall i\in\ then the formula simplifies to :f(t)=\mathcal^\left\


An example

We want to solve :f''(t)+4f(t)=\sin(2t) with initial conditions ''f''(0) = 0 and ''f′''(0)=0. We note that :\phi(t)=\sin(2t) and we get :\mathcal\=\frac The equation is then equivalent to :s^2\mathcal\-sf(0)-f'(0)+4\mathcal\=\mathcal\ We deduce :\mathcal\=\frac Now we apply the Laplace inverse transform to get :f(t)=\frac\sin(2t)-\frac\cos(2t)


Bibliography

* A. D. Polyanin, ''Handbook of Linear Partial Differential Equations for Engineers and Scientists'', Chapman & Hall/CRC Press, Boca Raton, 2002. {{isbn, 1-58488-299-9 Integral transforms Differential equations Differential calculus
Ordinary differential equations In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...