Glossary

6.2 Solving initial value problems using Laplace transforms

4 min read•august 15, 2024

Laplace transforms are powerful tools for solving initial value problems in differential equations. They convert complex time-domain equations into simpler algebraic ones in the frequency domain, making solutions easier to find.

Once transformed, these problems can be solved using basic algebra. The then brings the solution back to the time domain. This method is especially useful for systems with complicated inputs or initial conditions.

Initial Value Problems with Laplace Transforms

Transforming Functions and Derivatives

Top images from around the web for Transforming Functions and Derivatives

  • Laplace Transform for a trig function - Mathematics Stack Exchange View original

    Is this image relevant?

  • How do you find the Inverse Laplace transformation for a product of $2$ functions? - Mathematics ... View original

    Is this image relevant?

  • Laplace Transform for a trig function - Mathematics Stack Exchange View original

    Is this image relevant?

1 of 3

Top images from around the web for Transforming Functions and Derivatives

  • Laplace Transform for a trig function - Mathematics Stack Exchange View original

    Is this image relevant?

  • How do you find the Inverse Laplace transformation for a product of $2$ functions? - Mathematics ... View original

    Is this image relevant?

  • Laplace Transform for a trig function - Mathematics Stack Exchange View original

    Is this image relevant?

1 of 3
  • converts time-domain function f(t) into frequency-domain function F(s) using integral formula L{f(t)}=∫0∞e−stf(t)dtL\{f(t)\} = \int_0^\infty e^{-st}f(t)dt
  • Transform derivatives following rule L{f′(t)}=sF(s)−f(0)L\{f'(t)\} = sF(s) - f(0) where f(0) represents initial condition
  • Higher-order derivatives transformed using L{f(n)(t)}=snF(s)−sn−1f(0)−sn−2f′(0)−...−f(n−1)(0)L\{f^{(n)}(t)\} = s^n F(s) - s^{n-1}f(0) - s^{n-2}f'(0) - ... - f^{(n-1)}(0)
  • Initial conditions directly incorporated into transformed equation simplifies problem-solving process
  • property allows transformation of complex differential equations term by term
    • Example: For equation ad2ydt2+bdydt+cy=f(t)a\frac{d^2y}{dt^2} + b\frac{dy}{dt} + cy = f(t), transform each term separately

Setting Up IVPs in s-Domain

  • Initial value problems (IVPs) for linear differential equations converted to algebraic equations in
  • Process involves applying Laplace transform to both sides of differential equation
  • Include initial conditions in transformed equation
    • Example: For second-order IVP y′′+3y′+2y=e−t,y(0)=1,y′(0)=0y'' + 3y' + 2y = e^{-t}, y(0) = 1, y'(0) = 0
    • Transformed equation s2Y(s)−sy(0)−y′(0)+3(sY(s)−y(0))+2Y(s)=1s+1s^2Y(s) - sy(0) - y'(0) + 3(sY(s) - y(0)) + 2Y(s) = \frac{1}{s+1}
    • Simplified s2Y(s)+3sY(s)+2Y(s)=1s+1+s+3s^2Y(s) + 3sY(s) + 2Y(s) = \frac{1}{s+1} + s + 3
  • Resulting algebraic equation solved for Y(s) using standard algebraic techniques

Solving Linear Differential Equations

Algebraic Techniques in s-Domain

  • Solve transformed equation for F(s) using algebraic methods (factoring, combining like terms)
  • breaks down complex rational expressions into simpler terms
    • Example: 2s+5s2+4s+3=As+1+Bs+3\frac{2s+5}{s^2+4s+3} = \frac{A}{s+1} + \frac{B}{s+3}
  • finds numerators of partial fractions
    • For above example, solve system of equations: 2s+5=A(s+3)+B(s+1)2s+5 = A(s+3) + B(s+1)
  • applied to solve equations with function products in time domain
    • L{f(t)∗g(t)}=F(s)G(s)L\{f(t) * g(t)\} = F(s)G(s), where * denotes

Special Functions and Techniques

  • (Heaviside function) has Laplace transform L{u(t−a)}=e−assL\{u(t-a)\} = \frac{e^{-as}}{s}
  • transformed as L{δ(t−a)}=e−asL\{\delta(t-a)\} = e^{-as}
  • These special functions used in solving certain types of problems (discontinuous inputs, impulse responses)
  • Final step involves applying inverse Laplace transform to F(s) to obtain f(t)
    • Example: Solve y′′+2y′+y=u(t−2)y'' + 2y' + y = u(t-2) with y(0) = 0, y'(0) = 1
    • Transformed equation s2Y(s)+2sY(s)+Y(s)=e−2sss^2Y(s) + 2sY(s) + Y(s) = \frac{e^{-2s}}{s}
    • Solve for Y(s) and apply inverse transform

Inverse Laplace Transforms for Solutions

Techniques for Inverse Transforms

  • Inverse Laplace transform L^(-1) converts s-domain solution F(s) back to time-domain solution f(t)
  • Table of common Laplace transform pairs essential for efficient inversion
  • Linearity property allows term-by-term inversion of complex expressions
    • Example: L−1{3F(s)+2G(s)}=3f(t)+2g(t)L^{-1}\{3F(s) + 2G(s)\} = 3f(t) + 2g(t)
  • Shifting property L−1{F(s−a)}=eatf(t)L^{-1}\{F(s-a)\} = e^{at}f(t) crucial for handling exponential terms in s-domain
  • Partial fraction decomposition applied to rational functions not directly found in transform tables
    • Example: L−1{2s+5s2+4s+3}=L−1{1s+1+1s+3}=e−t+e−3tL^{-1}\{\frac{2s+5}{s^2+4s+3}\} = L^{-1}\{\frac{1}{s+1} + \frac{1}{s+3}\} = e^{-t} + e^{-3t}

Advanced Inversion Methods

  • Convolution theorem used for inverse transforms of products in s-domain
    • Results in convolutions in time domain L−1{F(s)G(s)}=f(t)∗g(t)L^{-1}\{F(s)G(s)\} = f(t) * g(t)
  • Contour integration techniques from complex analysis necessary for inverse transforms of complex expressions
    • Example: Bromwich integral for inverting transforms not found in standard tables
  • For solutions involving periodic functions, use of complex exponentials and Euler's formula may be necessary
    • Example: Inverse transform of ss2+ω2\frac{s}{s^2+\omega^2} yields cos⁡(ωt)\cos(\omega t)

Interpreting Solutions in Context

Analyzing Solution Behavior

  • Time-domain solution f(t) represents system behavior described by original differential equation
  • Initial conditions automatically satisfied due to incorporation in Laplace transform process
  • Long-term behavior analyzed using final value theorem of Laplace transforms
    • lim⁡t→∞f(t)=lim⁡s→0sF(s)\lim_{t \to \infty} f(t) = \lim_{s \to 0} sF(s) (if limit exists)
  • Identify and interpret transient and steady-state components of solution
    • Example: In solution y(t)=2e−t+3sin⁡(2t)y(t) = 2e^{-t} + 3\sin(2t), 2e−t2e^{-t} is transient, 3sin⁡(2t)3\sin(2t) is steady-state

Verifying and Visualizing Solutions

  • Singularities in F(s) correspond to important time-domain solution characteristics (oscillations, exponential growth/decay)
  • Verify solution by substituting back into original differential equation and checking initial conditions
  • Create graphical representations of solution for insights into system behavior over time
    • Example: Plot solution of damped harmonic oscillator to visualize decay and oscillation
  • Interpret solution in terms of physical system parameters and initial conditions
    • Example: In spring-mass system, relate solution coefficients to mass, spring constant, and damping factor

Key Terms to Review (23)

Boundary value condition: A boundary value condition is a set of constraints that specify the values a solution to a differential equation must take on the boundaries of its domain. These conditions are crucial in defining the behavior of solutions to partial differential equations, especially when applying methods like Laplace transforms to solve initial value problems. Understanding how these conditions interact with the governing equations is essential for finding unique and stable solutions.
Circuit analysis: Circuit analysis is the process of determining the voltage, current, and power in electrical circuits. It involves using various techniques to solve for unknown values within a circuit, which can include resistors, capacitors, and inductors. This analysis is crucial in understanding how electrical components interact and respond to different inputs, especially when applied to real-world systems.
Control Theory: Control theory is a mathematical framework used to analyze and design systems that can be controlled to achieve desired behaviors. It focuses on how to manipulate the inputs of a system to produce the desired output while maintaining stability and performance. This concept is crucial when considering the dynamics of systems, particularly in understanding stability analysis, solving initial value problems, and applying Laplace transforms for effective control design.
Convolution: Convolution is a mathematical operation that combines two functions to produce a third function, representing how one function influences the other. This operation plays a crucial role in solving differential equations and is particularly useful in transforming functions within integral equations. In the context of transforming and solving problems, convolution helps to express the output of linear systems and provides a way to handle initial value problems and partial differential equations efficiently.
Convolution Theorem: The Convolution Theorem states that the convolution of two functions in the time domain corresponds to the multiplication of their transforms in the frequency domain. This theorem is crucial for analyzing linear systems, as it simplifies the process of solving differential equations and integral equations by transforming convolutions into algebraic operations.
Dirac Delta Function: The Dirac delta function is a mathematical construct that represents a distribution rather than a traditional function, often denoted as \( \delta(x) \). It is defined to be zero everywhere except at the origin, where it is infinitely high, yet integrates to one over the entire real line. This unique property makes it extremely useful in modeling idealized point sources or instantaneous impulses in various mathematical and engineering contexts, particularly when dealing with discontinuous forcing terms and initial value problems using Laplace transforms.
Exponential Function: An exponential function is a mathematical function of the form $$f(t) = a e^{bt}$$, where 'a' is a constant, 'e' is the base of natural logarithms, and 'b' is the growth or decay rate. These functions are important in modeling processes that change at a rate proportional to their current value, making them useful in many applications, including solving initial value problems.
Homogeneous solution: A homogeneous solution refers to a specific type of solution to a differential equation where all terms are dependent solely on the function itself and its derivatives, with no additional forcing terms. This concept is crucial in understanding how to break down complex problems into simpler parts, allowing for the analysis of systems without external influences.
Initial value condition: An initial value condition refers to the specification of the value of a function and possibly its derivatives at a particular point, typically the starting point of the independent variable. This is crucial for solving differential equations, as it provides the necessary information to determine a unique solution that satisfies both the equation and these initial constraints.
Inverse Laplace Transform: The inverse Laplace transform is a mathematical operation that takes a function in the Laplace transform domain and converts it back into a function of time. This process is crucial for solving differential equations and initial value problems, as it allows for the recovery of time-domain solutions from their transformed counterparts. By applying properties such as linearity and convolution, the inverse Laplace transform can facilitate more complex analyses, including the use of Duhamel's principle for non-homogeneous problems.
Laplace Transform: The Laplace Transform is a mathematical operation that transforms a function of time into a function of a complex variable, typically denoted as 's'. It is particularly useful for solving differential equations and analyzing linear systems, allowing us to convert problems in the time domain into the frequency domain. This transformation simplifies the process of solving initial value problems and provides insights into system behavior through poles and zeros in the complex plane.
Linear ordinary differential equations: Linear ordinary differential equations (ODEs) are equations that involve an unknown function and its derivatives, where the unknown function and its derivatives appear linearly. This means that the equation can be expressed in the form $$a_n(t)y^{(n)} + a_{n-1}(t)y^{(n-1)} + ... + a_1(t)y' + a_0(t)y = g(t)$$, where the coefficients $$a_i(t)$$ are functions of the independent variable, and $$g(t)$$ is a known function. These equations are crucial for modeling many physical systems and can be solved using various methods, including the Laplace transform technique.
Linearity: Linearity refers to a property of equations or systems where the output is directly proportional to the input, meaning that if you scale the input, the output scales by the same factor. This concept is crucial in understanding how solutions to differential equations can be combined, leading to the superposition principle, which states that the sum of two solutions is also a solution. Linearity underpins many mathematical techniques, allowing for simplified analysis and manipulation of complex problems.
Method of undetermined coefficients: The method of undetermined coefficients is a technique used to find particular solutions of linear differential equations with constant coefficients by guessing a form of the solution and determining the coefficients through substitution. This method is particularly effective when the non-homogeneous term is a polynomial, exponential, sine, or cosine function. The goal is to express the general solution as a combination of the complementary solution and the particular solution derived from this guessing method.
Partial Differential Equations: Partial differential equations (PDEs) are equations that involve the partial derivatives of a function with respect to multiple variables. They are fundamental in describing various physical phenomena, such as heat conduction, fluid flow, and wave propagation. PDEs play a crucial role in solving initial value problems where the state of a system is determined by conditions at a specific point in time, making them essential for mathematical modeling across multiple disciplines.
Partial fraction decomposition: Partial fraction decomposition is a technique used to express a rational function as the sum of simpler fractions, allowing for easier integration and manipulation in mathematical problems. This method is particularly useful in the context of Laplace transforms, as it simplifies complex expressions into more manageable forms, making it easier to find inverse transforms or solve initial value problems.
Particular solution: A particular solution is a specific solution to a differential equation that satisfies both the equation itself and the initial or boundary conditions imposed on the problem. This type of solution represents a unique scenario within a family of solutions defined by the general solution, making it essential for solving initial value problems, applying principles to inhomogeneous equations, and addressing first-order PDEs.
Residuals: Residuals are the differences between the actual values and the estimated values produced by a mathematical model, reflecting how well the model fits the data. In the context of using Laplace transforms to solve initial value problems, residuals help determine the accuracy of the solution by indicating how much of the original problem's behavior is captured in the transformed domain. They are essential for evaluating and improving the effectiveness of the solution approach.
S-domain: The s-domain is a complex frequency domain used in the analysis of linear time-invariant systems, particularly in the context of Laplace transforms. It allows for the conversion of differential equations into algebraic equations by transforming functions from the time domain into the s-domain, where 's' represents a complex variable. This transformation simplifies the process of solving initial value problems by enabling the use of algebraic methods to find solutions in a straightforward manner.
Shifting Theorem: The Shifting Theorem is a fundamental property of the Laplace transform that describes how a shift in the time domain affects the transform in the frequency domain. Specifically, it states that if a function is shifted by a constant amount in time, the Laplace transform of the shifted function can be expressed in terms of the original function's transform, modified by an exponential factor. This theorem is particularly useful for solving initial value problems where the input function may have time shifts.
Step Function: A step function is a piecewise constant function that jumps from one value to another, often used to model situations with abrupt changes or discontinuities. It plays a crucial role in analyzing systems subjected to sudden forces or influences, as seen in the Heaviside function which serves as a common example of a step function. Understanding step functions is essential for solving initial value problems where external inputs change over time in a non-continuous manner.
T-domain: The t-domain refers to the time domain in which a function is defined, particularly in the context of differential equations and signal processing. It represents how a function or signal evolves over time, allowing for the analysis of initial conditions and temporal behavior of systems. Understanding the t-domain is crucial for solving initial value problems using Laplace transforms, as it establishes the basis for transforming functions into the s-domain for easier manipulation and analysis.
Unit Step Function: The unit step function, often denoted as $u(t)$, is a piecewise function that equals 0 for $t < 0$ and 1 for $t \geq 0$. This function is crucial in transforming initial value problems into the Laplace domain by allowing the inclusion of piecewise constant inputs and time-dependent conditions, making it easier to analyze systems with sudden changes or discontinuities.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide