Reduction of order in solving second order differential equations

[chwala]

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.

 

[renormalize]

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?

 

[chwala]

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.

 

[renormalize]

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.