Explanation on how these solutions can be easily seen

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SUMMARY

The discussion centers on demonstrating that the functions x1(t) = cos(t) and x2(t) = sin(t) are solutions to the differential equation (d²x/dt²) + x = 0. Participants emphasize the importance of calculating the second derivatives of these trigonometric functions to validate their solutions. The consensus is that once the second derivatives are computed, the relationship to the differential equation becomes clear and straightforward.

PREREQUISITES
  • Understanding of basic calculus, specifically differentiation.
  • Familiarity with trigonometric functions, particularly sine and cosine.
  • Knowledge of differential equations and their standard forms.
  • Ability to compute second derivatives of functions.
NEXT STEPS
  • Learn how to compute second derivatives of trigonometric functions.
  • Study the characteristics of linear differential equations.
  • Explore the general solutions to second-order differential equations.
  • Investigate the applications of trigonometric functions in solving differential equations.
USEFUL FOR

Students studying calculus, particularly those focusing on differential equations, as well as educators seeking to clarify concepts related to trigonometric solutions in mathematical contexts.

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Explanation on how these solutions can be "easily" seen

Homework Statement



Show that x1(t)=cos(t) and x2(t)=sin(t) are both solutions to the following Equation.

Homework Equations



(d2x/dt2) + x = 0


The Attempt at a Solution



Ok, So I just cannot understand how this can easily be seen! I cannot see it at all!
I know I haven't got any work to show you and that is wrong but I honestly do not know where to start with this.

It is in the middle of some lecture notes and all they say is "this can easily be seen"

Please Help this dummy!:'(
 
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Find the second derivatives of sin(t) and cos(t). Once you do that it should be 'easily seen'.
 


Gosh now I feel super dumb :P Thanks a million!
 

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