# Laplace transform applied to differential equations

The use of Laplace transform makes it much easier to solve linear differential equations with given initial conditions.

First consider the following relations:

[itex]\mathcal{L}\{f'\}
 = s \mathcal{L}\{f\} - f(0)[itex]

[itex]\mathcal{L}\{f''\}
 = s^2 \mathcal{L}\{f\} - s f(0) - f'(0)[itex]

[itex]\mathcal{L}\{f^{(n)}\}
 = s^n \mathcal{L}\{f\} - \Sigma_{i = 1}^{n}s^{n - i}f^{(i - 1)}(0)[itex]


Suppose we want to solve the given differential equation:

[itex]\sum^n_{i=0}a_if^{(i)}(t)=\phi(t)[itex]

This equation is equivalent to

[itex]\sum^n_{i=0}a_i\mathcal{L}\{f^{(i)}(t)\}=\mathcal{L}\{\phi(t)\}[itex]

which is equivalent to

[itex]\mathcal{L}\{f(t)\}={\mathcal{L}\{\phi(t)\}+\sum^n_{i=0}a_i\sum^i_{j=0}s^{i-j}f^{(j-i)}(0) \over \sum^n_{i=0}a_is^i}[itex]

note that the [itex]f^{(k)}(0)[itex] are initial conditions.

Then all we need to get f(t) is to apply the Laplace inverse transform to [itex]\mathcal{L}\{f(t)\}[itex]

## An example

We want to solve :

[itex]f^{(2)}(t)+4f(t)=\sin(2t) \,\![itex]

with initial conditions f(0) = 0 and f ′(0)=0

we note :

[itex]\phi(t)=\sin(2t) \,\![itex]

and we get :

[itex]\mathcal{L}\{\phi(t)\}=\frac{2}{s^2+4}[itex]

so this is equivalent to :

[itex]s^2\mathcal{L}\{f(t)\}-sf(0)-f^{(1)}(0)+4\mathcal{L}\{f(t)\}=\mathcal{L}\{\phi(t)\}[itex]

we deduce :

[itex]\mathcal{L}\{f(t)\}=\frac{2}{(s^2+4)^2}[itex]

So we apply the Laplace inverse transform and get

[itex]f(t)=\frac{1}{8}\sin(2t)-\frac{t}{4}\cos(2t) [itex]

## Bibliography

• A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, 2002.
##### Navigation

Academic Kids Menu

• Art and Cultures
• Art (http://www.academickids.com/encyclopedia/index.php/Art)
• Architecture (http://www.academickids.com/encyclopedia/index.php/Architecture)
• Cultures (http://www.academickids.com/encyclopedia/index.php/Cultures)
• Music (http://www.academickids.com/encyclopedia/index.php/Music)
• Musical Instruments (http://academickids.com/encyclopedia/index.php/List_of_musical_instruments)
• Biographies (http://www.academickids.com/encyclopedia/index.php/Biographies)
• Clipart (http://www.academickids.com/encyclopedia/index.php/Clipart)
• Geography (http://www.academickids.com/encyclopedia/index.php/Geography)
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
• Maps (http://www.academickids.com/encyclopedia/index.php/Maps)
• Flags (http://www.academickids.com/encyclopedia/index.php/Flags)
• Continents (http://www.academickids.com/encyclopedia/index.php/Continents)
• History (http://www.academickids.com/encyclopedia/index.php/History)
• Ancient Civilizations (http://www.academickids.com/encyclopedia/index.php/Ancient_Civilizations)
• Industrial Revolution (http://www.academickids.com/encyclopedia/index.php/Industrial_Revolution)
• Middle Ages (http://www.academickids.com/encyclopedia/index.php/Middle_Ages)
• Prehistory (http://www.academickids.com/encyclopedia/index.php/Prehistory)
• Renaissance (http://www.academickids.com/encyclopedia/index.php/Renaissance)
• Timelines (http://www.academickids.com/encyclopedia/index.php/Timelines)
• United States (http://www.academickids.com/encyclopedia/index.php/United_States)
• Wars (http://www.academickids.com/encyclopedia/index.php/Wars)
• World History (http://www.academickids.com/encyclopedia/index.php/History_of_the_world)
• Human Body (http://www.academickids.com/encyclopedia/index.php/Human_Body)
• Mathematics (http://www.academickids.com/encyclopedia/index.php/Mathematics)
• Reference (http://www.academickids.com/encyclopedia/index.php/Reference)
• Science (http://www.academickids.com/encyclopedia/index.php/Science)
• Animals (http://www.academickids.com/encyclopedia/index.php/Animals)
• Aviation (http://www.academickids.com/encyclopedia/index.php/Aviation)
• Dinosaurs (http://www.academickids.com/encyclopedia/index.php/Dinosaurs)
• Earth (http://www.academickids.com/encyclopedia/index.php/Earth)
• Inventions (http://www.academickids.com/encyclopedia/index.php/Inventions)
• Physical Science (http://www.academickids.com/encyclopedia/index.php/Physical_Science)
• Plants (http://www.academickids.com/encyclopedia/index.php/Plants)
• Scientists (http://www.academickids.com/encyclopedia/index.php/Scientists)
• Social Studies (http://www.academickids.com/encyclopedia/index.php/Social_Studies)
• Anthropology (http://www.academickids.com/encyclopedia/index.php/Anthropology)
• Economics (http://www.academickids.com/encyclopedia/index.php/Economics)
• Government (http://www.academickids.com/encyclopedia/index.php/Government)
• Religion (http://www.academickids.com/encyclopedia/index.php/Religion)
• Holidays (http://www.academickids.com/encyclopedia/index.php/Holidays)
• Space and Astronomy
• Solar System (http://www.academickids.com/encyclopedia/index.php/Solar_System)
• Planets (http://www.academickids.com/encyclopedia/index.php/Planets)
• Sports (http://www.academickids.com/encyclopedia/index.php/Sports)
• Timelines (http://www.academickids.com/encyclopedia/index.php/Timelines)
• Weather (http://www.academickids.com/encyclopedia/index.php/Weather)
• US States (http://www.academickids.com/encyclopedia/index.php/US_States)

Information

• Home Page (http://academickids.com/encyclopedia/index.php)
• Contact Us (http://www.academickids.com/encyclopedia/index.php/Contactus)

• Clip Art (http://classroomclipart.com)