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 ... ( ) 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 ), ... ( ) where a , b ∈ R .
When this function and its derivative are substituted in equation ( ), L.H.S. = R.H.S.. So it is a solution of the differential equation ( ). Let a and b be given some particular values say a = and b π , then we get a function y = φ ( x ) = 2sin π ...
( ) When this function and its derivative are substituted in equation ( ) again L.H.S. = R.H.S.. Therefore φ is also a solution of equation ( ). Function φ consists of two arbitrary constants (parameters) a , b and it is called general solution of the given differential equation.
Whereas function φ 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. Example Verify that the function y = e – x is a solution of the differential equation Solution Given function is y = e – x .
Differentiating both sides of equation with respect to