If you need a review on how to use this table do not forget to check the lesson on calculating Laplace transforms.īut the most important part of the process of solving differential equations with a Laplace transform comes from a derivation of equation 1 shown in the last section, this is, when we apply the Laplace transform formula to the derivative of y.įrom the definition of Laplace transform given in equation 1, you can see that the represented transformation is of a function of t, this is because we usually define y=f(t) or "the y function is in terms of t, and therefore is a function of t". Such table containing common Laplace transforms will allow us to once again solve difficult problems by comparison with the equations contained on the table. This process will be aided by the Laplace transform table shown below: In reality, we need both Laplace transforms and Inverse Laplace transforms in order to find the solution to an ordinary differential equation, the trick is to apply one first (which will allow us to change the differential equation to an expression containing only y's), simplify the equation as much as possible and then reverse it by taking the inverse transformation to solve to y.
Now is time to see how these transformations are helpful to us while solving differential equations. So we have already had an introduction to the Laplace transform and even a lesson on how to calculate Laplace expressions by a simple method of comparison. Solve differential equation using Laplace transform: Afterwards we will explain the calculations in a list of steps and we will finish by solving a few examples on the topic. In order to calculate such result, we will first compute the two main equations that will be used throughout the process, these equations which we recommend to learn and keep them at hand, are the ones shown in equation 6. This is exactly the case for our lesson of today, where we will use the Laplace transformation in order to decompose a higher order linear differential equation, separate its terms, simplify, and then work them through to obtain an expression for the implicit solution of the differential equation. It has been mentioned before on the introductory lesson, that mathematically speaking, we use the term transformations when referring to clever tricks in math that will allows to change a problem from higher level methodology into something simpler, such as algebra. Laplace transform for a function f(t) where t>=0. In addition, it solves higher-order equations with methods like undetermined coefficients, variation of parameters, the method of Laplace transforms, and many more.Equation 1. This step-by-step program has the ability to solve many types of first-order equations such as separable, linear, Bernoulli, exact, and homogeneous. What about equations that can be solved by Laplace transforms? Not a problem for Wolfram|Alpha:
Even differential equations that are solved with initial conditions are easy to compute. Get step-by-step directions on solving exact equations or get help on solving higher-order equations. Wolfram|Alpha can help out in many different cases when it comes to differential equations. Wolfram|Alpha can show the steps to solve simple differential equations as well as slightly more complicated ones like this one: Differential equations are fundamental to many fields, with applications such as describing spring-mass systems and circuits and modeling control systems.įrom basic separable equations to solving with Laplace transforms, Wolfram|Alpha is a great way to guide yourself through a tough differential equation problem. Today we’re pleased to introduce a new member to this family: step-by-step differential equations. Wolfram|Alpha has become well-known for its ability to perform step-by-step math in a variety of areas.