General and Particular Solutions of a Differential Equation

In earlier Classes, we have solved the equations of the type: 
`x^2` + 1 = 0              ... (1)
`sin^2 x` – cos x = 0 ... (2)
Solution of equations (1) and (2) are numbers, real or complex, that will satisfy the given equation i.e., when that number is substituted for the unknown x in the given equation, L.H.S. becomes equal to the R.H.S..  
Now consider the differential equation  
`(d^2y)/(dx^2) + y = 0`    ...(3)
In contrast to the first two equations, the solution of this differential equation is a function φ that will satisfy it i.e., when the function φ is substituted for the unknown y (dependent variable) in the given differential equation, L.H.S. becomes equal to R.H.S.. 
The curve y = φ (x) is called the solution curve (integral curve) of the given differential equation. Consider the function given by 
y = φ (x) = a sin (x + b), ... (4) 
where a, b ∈ R. When this function and its derivative are substituted in equation (3), L.H.S. = R.H.S.. 
So it is a solution of the differential equation (3).
Function φ consists of two arbitrary constants (parameters) a, b and it is called general solution of the given differential equation. Whereas function φ1 contains no arbitrary constants but only the particular values of the parameters a and b and hence is called a particular solution of the given differential equation.
The solution which contains  arbitrary constants is called the general solution (primitive) of the differential equation. 
The solution free from arbitrary constants i.e., the solution obtained from the general solution by giving particular values to the arbitrary constants is called a particular solution of the differential equation.