Graduate Applying the Laplace transform to solve Differential equations

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SUMMARY

The discussion centers on the application of the Laplace transform to finite-order differential equations, specifically addressing whether it can yield an infinite-order differential equation. It is established that the Laplace transform converts a differential equation into an algebraic equation, particularly effective for equations with constant coefficients. An example provided is the second-order differential equation ##y''(t)+\sin(t)y(t)=0##, which raises questions about the validity and purpose of obtaining an infinite-order equation through this method.

PREREQUISITES
  • Understanding of Laplace transforms and their properties
  • Familiarity with differential equations, particularly second-order equations
  • Knowledge of algebraic manipulation in the context of transforms
  • Concept of series expansion in mathematical analysis
NEXT STEPS
  • Study the properties of the Laplace transform in detail
  • Explore the implications of applying Laplace transforms to non-constant coefficient differential equations
  • Investigate the series expansion of trigonometric functions and their transforms
  • Learn about the relationship between finite and infinite-order differential equations
USEFUL FOR

Mathematicians, engineering students, and professionals working with differential equations and transforms, particularly those interested in advanced mathematical techniques for solving complex equations.

LagrangeEuler
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Is it possible to apply Laplace transform to some equation of finite order, second for instance, and get the differential equation of infinite order?
 
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A laplace transform turns a differential equation into an algebraic equation.
 
If you had a term like ##\sin t\,y##, you could expand ##\sin t## as a series and take the Laplace transform of the result term by term, which would give you a bunch of derivatives of Y(s). I'm not sure why you'd want to do that though or if doing so is valid.
 
pasmith said:
A laplace transform turns a differential equation into an algebraic equation.
It is only in the case when you have a differential equation with constant coefficients.
 
Do you have an example in mind?
 
##y''(t)+\sin(t)y(t)=0##, where ##y(0)=A##, ##y'(0)=B##. If I apply the Laplace transform would I get the differential equation of infinite order?
 

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