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:
Reduction of order involves assuming a solution of the form y = v(x)u(x), where v(x) is a known function and u(x) is an unknown function to be determined. By substituting y = v(x)u(x) into the second order differential equation, we can reduce it to a first order differential equation in terms of u(x).
Reduction of order is typically used when one solution of the homogeneous second order differential equation is known. By assuming a second linearly independent solution in the form of y = v(x)u(x), we can reduce the order of the differential equation and find the second solution.
The steps involved in reduction of order include assuming a solution of the form y = v(x)u(x), substituting it into the differential equation, solving for u(x) by reducing the order of the differential equation, and finding the general solution by combining the known solution with the newly found solution.
Reduction of order is typically used for homogeneous second order differential equations, where the right-hand side of the equation is zero. For nonhomogeneous equations, other methods such as variation of parameters or undetermined coefficients are more appropriate.
Reduction of order can simplify the process of solving second order differential equations by reducing them to first order differential equations. This method is particularly useful when one solution is known, as it allows us to find a second linearly independent solution easily.