Differential equations true/false question

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SUMMARY

The statement "If y1 and y2 are solutions to y'' - y = 0, then c1y1 + c2y2 represents all solutions to the differential equation for all scalars c1 and c2" is true only if y1 and y2 are linearly independent solutions. If y1 and y2 are not linearly independent, such as when y1 = e^x and y2 = 2e^x, the statement is false. The key concept here is the linear independence of solutions to the differential equation.

PREREQUISITES
  • Understanding of linear independence in the context of differential equations
  • Familiarity with the second-order linear differential equation y'' - y = 0
  • Knowledge of the general solution form for linear differential equations
  • Basic concepts of scalars and their role in linear combinations
NEXT STEPS
  • Study the concept of linear independence in differential equations
  • Learn about the Wronskian determinant and its application in determining independence
  • Explore the general solution of second-order linear differential equations
  • Investigate specific examples of solutions to y'' - y = 0 and their linear combinations
USEFUL FOR

Students studying differential equations, mathematics educators, and anyone seeking to deepen their understanding of linear combinations and solutions to linear differential equations.

freshman2013
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Homework Statement


If y1 and y2 are solutions to y"-y = 0 then c1y1+c2y2 represent all solutions to the differential equation for all scalars c1 and c2


Homework Equations





The Attempt at a Solution


Basically, my TA's solutions to his worksheet said it was true, and I'm not sure if he implied that y1 and y2 are linearly independent solutions, since in that case it would definitely be true. I'm not sure if I got the vocab mixed up or something. (y1 can be e^x and y2 can be 2e^x and that would make my original question false, right?)
 
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freshman2013 said:

Homework Statement


If y1 and y2 are solutions to y"-y = 0 then c1y1+c2y2 represent all solutions to the differential equation for all scalars c1 and c2


Homework Equations





The Attempt at a Solution


Basically, my TA's solutions to his worksheet said it was true, and I'm not sure if he implied that y1 and y2 are linearly independent solutions, since in that case it would definitely be true. I'm not sure if I got the vocab mixed up or something. (y1 can be e^x and y2 can be 2e^x and that would make my original question false, right?)

You have it exactly right. Unless the two solutions are linearly independent the answer is "false".
 

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