mathematics

Laplace transform applied to differential equations

Laplace transform applied to differential equations is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Laplace transform applied to differential equations rather than just read about it. In short: In mathematics, the Laplace transform is a powerful integral transform used to switch a function from the time domain to the s-domain. The Laplace transform can be used in some cases to solve linear differential equations with given initial conditions.

Key takeaways

Laplace transform applied to differential equations belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Laplace transform applied to differential equations to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Laplace transform applied to differential equations from memory before moving on to harder problems.

Reference excerpt

In mathematics, the Laplace transform is a powerful integral transform used to switch a function from the time domain to the s-domain. The Laplace transform can be used in some cases to solve linear differential equations with given initial conditions.

Approach First consider the following property of the Laplace transform:

L { f ′ } = s L { f } − f ( 0 ) {\displaystyle {\mathcal {L}}\{f'\}=s{\mathcal {L}}\{f\}-f(0)}

L { f ″ } = s 2 L { f } − s f ( 0 ) − f ′ ( 0 ) {\displaystyle {\mathcal {L}}\{f''\}=s^{2}{\mathcal {L}}\{f\}-sf(0)-f'(0)}

One can prove by induction that

L { f ( n ) } = s n L { f } − ∑ i = 1 n s n − i f ( i − 1 ) ( 0 ) {\displaystyle {\mathcal {L}}\{f^{(n)}\}=s^{n}{\mathcal {L}}\{f\}-\sum _{i=1}^{n}s^{n-i}f^{(i-1)}(0)}

Now we consider the following differential equation:

∑ i = 0 n a i f ( i ) ( t ) = ϕ ( t ) {\displaystyle \sum _{i=0}^{n}a_{i}f^{(i)}(t)=\phi (t)}

with given initial conditions

f ( i ) ( 0 ) = c i {\displaystyle f^{(i)}(0)=c_{i}}

Using the linearity of the Laplace transform it is equivalent to rewrite the equation as

∑ i = 0 n a i L { f ( i ) ( t ) } = L { ϕ ( t ) } {\displaystyle \sum _{i=0}^{n}a_{i}{\mathcal {L}}\{f^{(i)}(t)\}={\mathcal {L}}\{\phi (t)\}}

obtaining

L { f ( t ) } ∑ i = 0 n a i s i − ∑ i = 1 n ∑ j = 1 i a i s i − j f ( j − 1 ) ( 0 ) = L { ϕ ( t ) } {\displaystyle {\mathcal {L}}\{f(t)\}\sum _{i=0}^{n}a_{i}s^{i}-\sum _{i=1}^{n}\sum _{j=1}^{i}a_{i}s^{i-j}f^{(j-1)}(0)={\mathcal {L}}\{\phi (t)\}}

Solving the equation for L { f ( t ) } {\displaystyle {\mathcal {L}}\{f(t)\}} and substituting f ( i ) ( 0 ) {\displaystyle f^{(i)}(0)} with c i {\displaystyle c_{i}} one obtains

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Laplace transform applied to differential equations

Start with the simplest possible case. Write down what Laplace transform applied to differential equations claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Laplace transform applied to differential equations before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Laplace transform applied to differential equations ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Laplace transform applied to differential equations

In research: Laplace transform applied to differential equations appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Laplace transform applied to differential equations in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Laplace transform applied to differential equations is common in secondary-school and first-year university syllabi. It links to neighbouring topics Differential calculus, Differential equations, Integral transforms, so understanding it makes those chapters shorter.
In everyday life: Look for Laplace transform applied to differential equations outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Laplace transform applied to differential equations in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Laplace transform applied to differential equations means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Laplace transform applied to differential equations out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Laplace transform applied to differential equations in simple terms?

Why does Laplace transform applied to differential equations matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Laplace transform applied to differential equations?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Laplace transform applied to differential equations.