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.