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Show that the Function is a Solution of the Differential Equation?

TopDifferential equation can be defined as equation which includes Derivatives. Differential equations may be of first or second or third order and are represented as shown below.
d x / d t – 2 x = 0 represents the first order differential equation, second order differential equation can be represented as:
d2 x / d t2 – 2 d x / d t + 3 x = 0.

Let’s consider a differential equation to show that the function is a solution of the differential equation as shown below.
Consider the following differential equation:
d2 x / d t2 + 2 (d x / d t) – 3x = 0,
And also consider two Functions as x1 (t) = e-3t and x2 (t) = et.
Since x1 (t) = e-3t, then differentiating x1 (t) = e-3t with respect to ‘t’ gives d x1 / d t = -3 e-3t, and similarly again differentiating d x1 / d t = -3 e-3t with respect to ‘t’ gives d2 x1 /d t2 = 9e-3t. Using the given equation and substituting the value of d2 x1 /d t2 and d x1 / d t results in d2 x1 / d t2 + 2 (d x1 / d t) – 3x1 = 9e-3t + 2 (-3 e-3t) – 3 e-3t = 9e-3t – 6 e-3t – 3 e-3t = 9e-3t – 9 e-3t = 0.
Hence it has been proved that function x1 (t) = e-3t proves the differential equation d2 x / d t2 + 2 (d x / d t) – 3x = 0,
Let’s take second function, x2 (t) = et,
d2 x/ d t2 = et.

Substituting the values of d x2 / d t and d2 x2 / d t2 in differential equation d2 x / d t2 + 2 (d x / d t) – 3x = 0, we get following result.
d2 x2 / d t2 + 2 (d x / d t) – 3x2 = et + 2et - 3et = 3et – 3et = 0, thus second function x2 (t) = et also satisfies the given differential equation.Differential equation can be defined as equation which includes derivatives. Differential equations may be of first or second or third order and are represented as shown below.
d x / d t – 2 x = 0 represents the first order differential equation, second order differential equation can be represented as:
d2 x / d t2 – 2 d x / d t + 3 x = 0.