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Solve initial value problem using laplace transform calculator

Step-by-step example for solving the initial value problem with a table of Laplace transforms

Example

Use a Laplace transform to solve the differential equation.

???y''-10y'+9y=5t???

with ???y(0)=-1??? and ???y'(0)=2???

To solve this problem using Laplace transforms, we will need to transform every term in our given differential equation. From a table of Laplace transforms, we can redefine each term in the differential equation.

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

???-10y'=-10\left[sY(s)-y(0)\right]???

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

???5t=\frac{5}{s^2}???

Plugging the transformed values back into the original equation gives

???s^2Y(s)-sy(0)-y'(0)-10\left[sY(s)-y(0)\right]+9Y(s)=\frac{5}{s^2}???

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

???s^2Y(s)-s(-1)-(2)-10\left[sY(s)-(-1)\right]+9Y(s)=\frac{5}{s^2}???

???s^2Y(s)+s-2-10\left[sY(s)+1\right]+9Y(s)=\frac{5}{s^2}???

???s^2Y(s)+s-2-10sY(s)-10+9Y(s)=\frac{5}{s^2}???

???s^2Y(s)+s-10sY(s)+9Y(s)-12=\frac{5}{s^2}???

From here we want to solve for ???Y(s)??? so that we can use a reverse Laplace transform to change this equation into an equation for ???y(t)???.

???s^2Y(s)-10sY(s)+9Y(s)=\frac{5}{s^2}+12-s???

???s^2Y(s)-10sY(s)+9Y(s)=\frac{5+12s^2-s^3}{s^2}???

???Y(s)\left(s^2-10s+9\right)=\frac{5+12s^2-s^3}{s^2}???

???Y(s)(s-9)(s-1)=\frac{5+12s^2-s^3}{s^2}???

???Y(s)(s-9)(s-1)=\frac{5+12s^2-s^3}{s^2}???

???Y(s)=\frac{5+12s^2-s^3}{s^2(s-9)(s-1)}???

We’ll need to use a partial fractions decomposition.

???\frac{5+12s^2-s^3}{s^2(s-9)(s-1)}=\frac{A}{s}+\frac{B}{s^2}+\frac{C}{s-9}+\frac{D}{s-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=As\left(s^2-10s+9\right)+B\left(s^2-10s+9\right)???

???+C\left(s^3-s^2\right)+D\left(s^3-9s^2\right)???

???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\right)+\left(-10As^2+Bs^2-Cs^2-9Ds^2\right)???

???+\left(9As-10Bs\right)+9B???

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

???+\left(9A-10B\right)s+9B???

Equating coefficients, we get a system of linear equations.

???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-10\left(\frac59\right)=0???

???9A-\frac{50}{9}=0???

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

???81A=50???

???A=\frac{50}{81}???

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

???\frac{50}{81}+C+D=-1???

???-10\left(\frac{50}{81}\right)+\frac59-C-9D=12???

which is

???C+D=-1-\frac{50}{81}???

???-\frac{500}{81}+\frac59-C-9D=12???

which is

???C+D=-\frac{81}{81}-\frac{50}{81}???

???-C-9D=12+\frac{500}{81}-\frac59???

which is

???C+D=-\frac{131}{81}???

???-C-9D=\frac{972}{81}+\frac{500}{81}-\frac{45}{81}???

which is

[1] ???C+D=-\frac{131}{81}???

[2] ???-C-9D=\frac{1,427}{81}???

Adding [1] and [2] together gives

???C+D+(-C-9D)=-\frac{131}{81}+\frac{1,427}{81}???

???C+D-C-9D=\frac{1,296}{81}???

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

???-8D=16???

[3] ???D=-2???

Plugging [3] into [1] we get

???C+D=-\frac{131}{81}???

???C-2=-\frac{131}{81}???

???C=-\frac{131}{81}+\frac{162}{81}???

???C=\frac{31}{81}???

Plugging the values we found for ???A???, ???B???, ???C??? and ???D??? back into the partial fractions decomposition will give us

???Y(s)=\frac{\frac{50}{81}}{s}+\frac{\frac{5}{9}}{s^2}+\frac{\frac{31}{81}}{s-9}-\frac{2}{s-1}???

We’ll rearrange each term in the decomposition to make it easier to find a matching formula in the Laplace transform table.

???Y(s)=\frac{50}{81}\left(\frac{1}{s}\right)+\frac{5}{9}\left(\frac{1}{s^2}\right)+\frac{31}{81}\left(\frac{1}{s-9}\right)-2\left(\frac{1}{s-1}\right)???

The terms remaining inside the parentheses should remind us of these transformations:

???1=\frac{1}{s}???

???t=\frac{1}{s^2}???

???e^{9t}=\frac{1}{s-9}???

???e^{t}=\frac{1}{s-1}???

We’ll make these substitutions to get an equation for ???y(t)???.

???y(t)=\frac{50}{81}+\frac{5}{9}t+\frac{31}{81}e^{9t}-2e^t???

This is the solution to the initial value problem.

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Intros

Lessons

  2. What is LL{yy'}? What is LL{yy''}? And how do we use these facts to calculate differential equations using Laplace Transforms?

  3. A brief run-down on the steps used to solve a differential equation by using the Laplace Transform.

Examples

Lessons

  1. Calculating Differential Equations Using Laplace Transforms

    Solve the initial value differential equation:

    y′′−3y′ +2y=6y'' - 3y' + 2y = 6

    With initial values yy(0) = 2 , y y'(0) = 6

    1. Solve the initial value differential equation:

      y′′−4y′+7y=0y'' - 4y' + 7y = 0

      With initial values yy(0) = 3, yy'(0) = 7

      How do you use Laplace transform to solve initial value problems?

      To use Laplace transform to solve initial value problem, a. Take the Laplace transform of both sides of the equation. b. Use the properties of the Laplace transform and the initial conditions to obtain an equation for the Laplace transform of the solution and then solve this equation for the transform.

      How do you find the initial value of Laplace transform?

      Taking differentiation on both sides, we get,.
      L[dx(t)dt]=∫∞0dx(t)dte−stdt..
      L[dx(t)dt]=∫∞0dx(t)dte−stdt=sX(s)−x(0−).
      lims→∞{∫∞0dx(t)dte−stdt}=lims→∞{sX(s)−x(0)}.

      How do you do Laplace transform on a calculator?

      The procedure to use the Laplace transform calculator is as follows:.
      Step 1: Enter the function, variable of function, transformation variable in the input field..
      Step 2: Click the button “Calculate” to get the integral transformation..
      Step 3: The result will be displayed in the new window..

      How do you solve a Laplace transform problem?

      The solution is accomplished in four steps:.
      Take the Laplace Transform of the differential equation. We use the derivative property as necessary (and in this case we also need the time delay property) ... .
      Put initial conditions into the resulting equation..
      Solve for Y(s).
      Get result from the Laplace Transform tables. (.

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