The discussion centers on the reduction of order technique in solving second-order differential equations, specifically addressing the treatment of constants in the general solution. Participants clarify that the constant \( c \) is not dropped but rather assigned a specific value, such as \( c = -3 \). The constant \( k \) is similarly treated as a redefinition of the arbitrary constant \( c_1 \). Ultimately, maintaining two arbitrary constants in the solution confirms that a general solution has been achieved.
PREREQUISITESMathematicians, engineering students, and anyone involved in solving second-order differential equations will benefit from this discussion, particularly those seeking clarity on the treatment of constants in their solutions.
Because keeping ##k## is the same thing as adjusting the value of the arbitrary constant ##c_1## in the general solution?chwala said:
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. .renormalize said:
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.chwala said: