# Solving Initial Value Problems Examples

The point of this problem however, was to show how we would use Laplace transforms to solve an IVP.

There are a couple of things to note here about using Laplace transforms to solve an IVP.

Combining the two terms gives, $Y\left( s \right) = \frac$ The partial fraction decomposition for this transform is, $Y\left( s \right) = \frac \frac \frac \frac$ Setting numerators equal gives, $5 12 - = As\left( \right)\left( \right) B\left( \right)\left( \right) C\left( \right) D\left( \right)$ Picking appropriate values of $$s$$ and solving for the constants gives, $\begin & s = 0 & 5 & = 9B & \Rightarrow \hspace B & = \frac\\ & s = 1 & 16 & = - 8D & \Rightarrow \hspace D & = - 2\\ & s = 9 & 248 & = 648C & \Rightarrow \hspace C & = \frac\\ & s = 2 & 45 & = - 14A \frac & \Rightarrow \hspace A & = \frac\end$ Plugging in the constants gives, $Y\left( s \right) = \frac \frac \frac - \frac$ Finally taking the inverse transform gives us the solution to the IVP.

$y\left( t \right) = \frac \fract \frac - 2$ That was a fair amount of work for a problem that probably could have been solved much quicker using the techniques from the previous chapter.

In many of the later problems Laplace transforms will make the problems significantly easier to work than if we had done the straight forward approach of the last chapter.

Also, as we will see, there are some differential equations that simply can’t be done using the techniques from the last chapter and so, in those cases, Laplace transforms will be our only solution.

Due to the nature of the mathematics on this site it is best views in landscape mode.

If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

First, using Laplace transforms reduces a differential equation down to an algebra problem.

In the case of the last example the algebra was probably more complicated than the straight forward approach from the last chapter. The algebra, while still very messy, will often be easier than a straight forward approach.

## Comments Solving Initial Value Problems Examples

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Initial Value Problems. When we solve differential equations, often times we will obtain many if not infinitely many solutions. For example, consider the.…

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The examples in this section are restricted to differential equations that. are often what we've been using for initial condition in our IVP's. That was a fair amount of work for a problem that probably could have been solved.…

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This problem can be solved in two steps. 1. 2. Using the initial data, plug it into the general solution and solve for c. EXAMPLE 1 Solve the initial value problem.…

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Initial Value Problems Example Question #1. If \displaystyle c is some constant and the initial value of the function, \displaystyle y=ce^x is six, determine the.…

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Initial Value Problem. An Initial Value Problem or IVP is a differential equation along with an appropriate number of initial conditions. Example.…

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See Section D.3.1 for an example of how this procedure can be used to. some region D then there is a unique solution to the initial value problem 5.14 at.…

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Initial Value Problems Examples. BACK · NEXT. Example 1. Is the function y = 4x + 1 a solution to the IVP. The function y = 4x + 1 satisfies the differential.…

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In the field of differential equations, an initial value problem is an ordinary. Second example. The solution of. y ′ + 3 y = 6 t + 5, y 0.…

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Free ebook A basic example showing how to solve an initial value problem involving a separable differential.…