Cosider the following equation
`color{orange} {x^2 +1 = 0}`.............................(1)
`color{orange} {sin^2x - 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 `color{orange} {(d^2y)/(dy^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
`color{orange} {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).
`=>` Let `a` and `b` be given some particular values say `a = 2` and `b = pi/4`, then we get a function
`color{orange} {y = y = φ_1(x) = 2sin (x + pi/4)}` .............................(5)
When this function and its derivative are substituted in equation (3) again L.H.S. = R.H.S.. Therefore φ1 is also a solution of equation (3).
`=>`The function `φ` consists of two arbitrary constants (parameters) `a, b` and it is called general solution of the given differential equation.
`=>` Whereas function `φ_1` 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.
`color{red}{=>"The solution which contains arbitrary constants is called the general solution (primitive) of the differential equation."}`
`color{red}{=>" The solution free from arbitrary constants i.e., the solution obtained from the general solution"}`
`color{red}{" by giving particular values to the arbitrary constants is called a particular solution of the differential equation."}`
Cosider the following equation
`color{orange} {x^2 +1 = 0}`.............................(1)
`color{orange} {sin^2x - 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 `color{orange} {(d^2y)/(dy^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
`color{orange} {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).
`=>` Let `a` and `b` be given some particular values say `a = 2` and `b = pi/4`, then we get a function
`color{orange} {y = y = φ_1(x) = 2sin (x + pi/4)}` .............................(5)
When this function and its derivative are substituted in equation (3) again L.H.S. = R.H.S.. Therefore φ1 is also a solution of equation (3).
`=>`The function `φ` consists of two arbitrary constants (parameters) `a, b` and it is called general solution of the given differential equation.
`=>` Whereas function `φ_1` 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.
`color{red}{=>"The solution which contains arbitrary constants is called the general solution (primitive) of the differential equation."}`
`color{red}{=>" The solution free from arbitrary constants i.e., the solution obtained from the general solution"}`
`color{red}{" by giving particular values to the arbitrary constants is called a particular solution of the differential equation."}`
Verify that the function `y = e^(- 3x)` is a solution of the differential equation `(d^y)/(dx^2) + (dy)/(dx) - 6y =0`
