General and Particular Solutions of differentail Equations
6. GENERAL AND PARTICULAR SOLUTIONS :
The solution of a differential equation which contains a number of independent arbitrary constants equal to the order of the differential equation is called the GENERAL SOLUTION (OR COMPLETE INTEGRAL OR COMPLETE PRIMITIVE). A solution obtainable from the general solution by giving particular values to the constants is called particular solution.
Note that the general solution of a differential equation of the nth order contains ‘n’ & only ‘n’ independent arbitrary constants. The arbitrary constants in the solution of a differential equation are said to be independent , when it is impossible to deduce from the solution an eqivalent relation containing fewer arbitrary constants. thus the two arbitrary constants A, B in the equation y = A ex + b are not incependent since the equation can be written as y = A eB . ex = Cex. Similarly the solution y = A sinx + B cos (x + C) appears to contain three arbitrary constants, but they are really equivalent to two only.
Similarly y = (C1 + C2) cos (x + C3) –C4 ex+c5 has only 3 independent arbitrary constants hence will be of order 3.
