**General and Particular Solution of a
Differential Equation**

We have solved the equation of the type:

x² + 1 = 0 … (1)

sin²x – cosx = 0 … (2)

solution of equation (1) and (2) are numbers, real or complex, that will satisfy the given equation i.e., when that number is substituted for the unknow x in the given equation, L.H.S. becomes equal to the R.H.S.

now consider the differential equation d²y/ dx² + y = 0 … (3)

the solution of this differential equation is a function φ that will satisfy it i.e., when the function φ is substituted for the unknow y 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 ϵ 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 = π/4, then we get a function

y = φ₁ (x) = 2 sin (x + π/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).

Function φ
consists of two arbitrary constant 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 hence is
called **a Particular solution of the given differential equation**.

**Example:** verify that the function y = e⁻³^{x}
is a solution of the differential equation \(\frac{{{d}^{2}}y}{d{{x}^{2}}}+\frac{dy}{dx}-6y=0\).

**Solution:
**Given that y = e⁻³^{x}

Differentiate with respect x

\(\frac{dy}{dx}=-3\times {{e}^{-3x}}\) … (1)

Again, Differentiate with respect x

\(\frac{{{d}^{2}}y}{d{{x}^{2}}}=-3\times -3\times {{e}^{-3x}}\),

\(\frac{{{d}^{2}}y}{d{{x}^{2}}}=9{{e}^{-3x}}\) … (2)

From equation (1) and (2)

\(\frac{{{d}^{2}}y}{d{{x}^{2}}}+\frac{dy}{dx}-6y=0\) (given that)

9e⁻³^{x}
– 3e⁻³^{x} – 6y = 0

9e⁻³^{x}
– 3e⁻³^{x} – 6e⁻³^{x} = 0

9e⁻³^{x}
– 9e⁻³^{x} = 0

L.H.S = R.H.S

Therefore, the given function is a solution of the given differential equation.