How to settle initial value difficulties making use of Laplace transforms

To use a Lalocation transform to deal with a second-order nonhomogeneous differential equations initial worth problem, we’ll have to use a table of Lalocation transcreates or the definition of the Laplace transcreate to put the differential equation in terms of ???Y(s)???.

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Once we settle the resulting equation for ???Y(s)???, we’ll desire to simplify it until we recognize that the terms in our equation match formulas in a table of Lalocation transcreates. Then we’ll make reverse substitutions for ???s??? in regards to ???t???.


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Given a differential equation and also initial problems, usage a table of Lalocation transcreates or the meaning to fix the initial value problem



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Step-by-step example for resolving the initial value trouble via a table of Laarea transforms

Example

Use a Laplace transform to deal with the differential equation.

???y""-10y"+9y=5t???

via ???y(0)=-1??? and also ???y"(0)=2???

To solve this difficulty utilizing Laarea transforms, we will should transform eincredibly term in our offered differential equation. From a table of Lalocation transforms, we have the right to redefine each term in the differential equation.

???y""=s^2Y(s)-sy(0)-y"(0)???

???-10y"=-10left???

???9y=9Y(s)???

???5t=frac5s^2???

Plugging the transformed values ago right into the original equation gives

???s^2Y(s)-sy(0)-y"(0)-10left+9Y(s)=frac5s^2???

Now we’ll plug in the given initial conditions ???y(0)=-1??? and ???y"(0)=2???.

???s^2Y(s)-s(-1)-(2)-10left+9Y(s)=frac5s^2???

???s^2Y(s)+s-2-10left+9Y(s)=frac5s^2???

???s^2Y(s)+s-2-10sY(s)-10+9Y(s)=frac5s^2???

???s^2Y(s)+s-10sY(s)+9Y(s)-12=frac5s^2???

From below we want to solve for ???Y(s)??? so that we can use a reverse Laarea transcreate to readjust this equation into an equation for ???y(t)???.

???s^2Y(s)-10sY(s)+9Y(s)=frac5s^2+12-s???

???s^2Y(s)-10sY(s)+9Y(s)=frac5+12s^2-s^3s^2???

???Y(s)left(s^2-10s+9 ight)=frac5+12s^2-s^3s^2???

???Y(s)(s-9)(s-1)=frac5+12s^2-s^3s^2???

???Y(s)(s-9)(s-1)=frac5+12s^2-s^3s^2???

???Y(s)=frac5+12s^2-s^3s^2(s-9)(s-1)???


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We’ll must usage a partial fractions decomplace.

???frac5+12s^2-s^3s^2(s-9)(s-1)=fracAs+fracBs^2+fracCs-9+fracDs-1???

???5+12s^2-s^3=As(s-9)(s-1)+B(s-9)(s-1)???

???+Cs^2(s-1)+Ds^2(s-9)???

???5+12s^2-s^3=Asleft(s^2-10s+9 ight)+Bleft(s^2-10s+9 ight)???

???+Cleft(s^3-s^2 ight)+Dleft(s^3-9s^2 ight)???

???5+12s^2-s^3=As^3-10As^2+9As+Bs^2-10Bs+9B???

???+Cs^3-Cs^2+Ds^3-9Ds^2???

???5+12s^2-s^3=left(As^3+Cs^3+Ds^3 ight)+left(-10As^2+Bs^2-Cs^2-9Ds^2 ight)???

???+left(9As-10Bs ight)+9B???

???5+12s^2-s^3=left(A+C+D ight)s^3+left(-10A+B-C-9D ight)s^2???

???+left(9A-10B ight)s+9B???

Equating coefficients, we gain a device of straight equations.

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???A+C+D=-1???

???-10A+B-C-9D=12???

???9A-10B=0???

???9B=5???

Solving the fourth equation for ???B??? gives

???B=frac59???

Plugging this into the third equation gives

???9A-10left(frac59 ight)=0???

???9A-frac509=0???

???81A-50=0???

???81A=50???

???A=frac5081???

Plugging the values we’ve uncovered for ???A??? and also ???B??? into the first two equation gives

???frac5081+C+D=-1???

???-10left(frac5081 ight)+frac59-C-9D=12???

which is

???C+D=-1-frac5081???

???-frac50081+frac59-C-9D=12???

which is

???C+D=-frac8181-frac5081???

???-C-9D=12+frac50081-frac59???

which is

???C+D=-frac13181???

???-C-9D=frac97281+frac50081-frac4581???

which is

<1>???C+D=-frac13181???

<2>???-C-9D=frac1,42781???

Adding<1>and<2>together gives

???C+D+(-C-9D)=-frac13181+frac1,42781???

???C+D-C-9D=frac1,29681???

???D-9D=16???

???-8D=16???

<3>???D=-2???

Plugging<3>into<1>we get

???C+D=-frac13181???

???C-2=-frac13181???

???C=-frac13181+frac16281???

???C=frac3181???

Plugging the values we discovered for ???A???, ???B???, ???C??? and also ???D??? earlier into the partial fractions decomposition will offer us

???Y(s)=fracfrac5081s+fracfrac59s^2+fracfrac3181s-9-frac2s-1???

We’ll rearrange each term in the decomplace to make it easier to find a equivalent formula in the Laplace transcreate table.

???Y(s)=frac5081left(frac1s ight)+frac59left(frac1s^2 ight)+frac3181left(frac1s-9 ight)-2left(frac1s-1 ight)???

The terms staying inside the parentheses need to remind us of these transformations: