I Reduction of order in solving second order differential equations

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TL;DR Summary
Why is the constant dropped when determining solutions to second order differential equations. (See highlight in red -attached). Otherwise, the reduction of order approach is pretty straightforward.
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chwala said:
TL;DR Summary: Why is the constant dropped when determining solutions to second order differential equations. (See highlight in red -attached). Otherwise, the reduction of order approach is pretty straightforward.

View attachment 338047
Because keeping ##k## is the same thing as adjusting the value of the arbitrary constant ##c_1## in the general solution?
 
renormalize said:
Because keeping ##k## is the same thing as adjusting the value of the arbitrary constant ##c_1## in the general solution?
Thanks noted...was wondering why they substituted for the constant ##c=-3## and dropped the other constant ##k##. The constant ##c## was not dropped as indicated rather a value was assigned to it. .

Is this not for convenience? to perhaps have " nice solutions'.

Cheers man.
 
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chwala said:
Thanks noted...was wondering why they substituted for the constants ##c=-3## and dropped ##k##. The constant ##c## was not dropped as indicated rather a value was assigned to it. .
Just like keeping ##k## amounts to redefining ##c_1##, wouldn't keeping ##-c/3## simply adjust the value of the arbitrary constant ##c_2##? These are the types of simplifications that come naturally once you're practiced enough solving differential questions. When solving a 2nd-order ODE, as long as you're left in the end with two arbitrary constants, you know you've found the general solution.
 
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