Verify that the function y = e^ -3x is a solution of the differen… - Vidyadip

Question

Verify that the function $y = e^{-3x}$ is a solution of the differential equation $\frac{d^{2} y}{d x^{2}}+\frac{d y}{d x} - 6y = 0 $

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Answer

Given function is $y = e^{-3x}$
Differentiating both sides of equation with respect to $x$ , we get
$\frac{d y}{d x}=-3 e^{-3x} ...(i)$
Now, differentiating $(1)$ with respect to $x$, we have$\frac{d^{2} y}{d x^{2}}=9 e^{-3 x}$
Substituting the values of $\frac{d^{2} y}{d x^{2}}, \frac{d y}{d x}$ and $y$ in the given differential equation, we get
$L.H.S. = 9.e^{-3x} + (-3.e^{-3x}) - 6.e^{-3x} = 9.e^{-3x} - 9.e^{-3x} = 0 = R.H.S.$
Therefore, the given function is a solution of the given differential equation.

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