# Laplace Transform Solved Problems

$\begin\mathcal\left\ & = s Y\left( s \right) - y\left( 0 \right)\\ \mathcal\left\ & = Y\left( s \right) - sy\left( 0 \right) - y'\left( 0 \right)\end$ Notice that the two function evaluations that appear in these formulas, $$y\left( 0 \right)$$ and $$y'\left( 0 \right)$$, are often what we’ve been using for initial condition in our IVP’s.

So, this means that if we are to use these formulas to solve an IVP we will need initial conditions at $$t = 0$$.

Now we're just taking Laplace Transforms, and let's see where this gets us. So I get the Laplace Transform of y-- and that's good because it's a pain to keep writing it over and over-- times s squared plus 5s plus 6. Because the characteristic equation to get that, we substituted e to the rt, and the Laplace Transform involves very similar function. What I'm going to do is I'm going to solve this.

And actually I just want to make clear, because I know it's very confusing, so I rewrote this part as this. I'm going to say the Laplace Transform of y is equal to something. We haven't solved for y yet, but we know that the Laplace Transform of y is equal to this.

We are trying to find the solution, $$y(t)$$, to an IVP.

What we’ve managed to find at this point is not the solution, but its Laplace transform.

Now, to use the Laplace Transform here, we essentially just take the Laplace Transform of both sides of this equation. So we get the Laplace Transform of y the second derivative, plus-- well we could say the Laplace Transform of 5 times y prime, but that's the same thing as 5 times the Laplace Transform-- y prime. I took this part and replaced it with what I have in parentheses.

So minus y prime of 0-- and now I'll switch colors-- plus 5 times-- once again the Laplace Transform of y prime. So 5 times s times Laplace Transform of y, minus y of 0, plus 6 times the Laplace Transform-- oh I ran out of space, I'll do it in another line-- plus 6 times the Laplace Transform of y. I know this looks really confusing but we'll simplify right now.

So let's scroll down a little bit, just so we have some breathing room. And it actually turns out it's a sum of things we already know, and we just have to manipulate this a little bit algebraically.

And so I get the Laplace Transform of y, times s squared, plus 5s, plus 6, is equal to-- let's add these terms to both sides of this equation-- is equal to 2s plus 3 plus 10-- oh, that's silly-- plus 13.

## Comments Laplace Transform Solved Problems

• ###### Laplace Transform to Solve a Differential Equation, Ex 1, Part 1/2.

Laplace Transform to Solve a Differential Equation, Ex 1, Part 1/2. In this video, I begin showing how to use the Laplace transform to solve a differential equation.…

• ###### Solns41 Chapter 4 Laplace transforms Solutions

Chapter 4 Laplace transforms Solutions The table of Laplace transforms is used throughout. we take the Laplace transform of both sides of the differential.…

• ###### Chapter 6 Laplace Transforms - 國立中正大學資工系

Laplace Transform The Laplace transform is a method of solving ODEs and initial value problems. The crucial idea is that operations of calculus on functions are replaced by operations of algebra on transforms. Roughly, differentiation of ft will correspond to multiplication of Lf by s see Theorems 1 and 2 and integration of…

• ###### Laplace Transform Practice Problems

Laplace Transform Practice Problems Answers on the last page A Continuous Examples no step functions Compute the Laplace transform of the given function.…

• ###### Some Additional Examples Laplace Transform

In addition to the Fourier transform and eigenfunction expansions, it is sometimes convenient to have the use of the Laplace transform for solving certain problems in partial differential equations. We will quickly develop a few properties of the Laplace transform and use them in solving some example problems. Laplace Transform…

• ###### Laplace transform Solved Problems 1 - Semnan University

LAPLACE TRANSFORM Many mathematical problems are solved using transformations. The idea is to transform the problem into another problem that is easier to solve. Once a solution is obtained, the inverse transform is used to obtain the solution to the original problem. The Laplace transform is an important tool that makes…

Further Studies of Laplace Transform 15 45 The Laplace Transform and the Method of Partial Fractions 28 46 Laplace Transforms of Periodic Functions 35 47 Convolution Integrals 45 48 The Dirac Delta Function and Impulse Response 53 49 Solving Systems of Di erential Equations Using Laplace Trans-form 61 50 Solutions to Problems 68 2…