Verify that the Function Y = E−3x is a Solution of the Differential Equation D 2 Y D X 2 + D Y D X − 6 Y = 0. - Mathematics

Sum

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

Solution

We have
$\frac{d^2 y}{d x^2} + \frac{dy}{dx} - 6y = 0 . . . . . \left( 1 \right)$
$Now,$
$y = e^{- 3x}$
$\Rightarrow \frac{dy}{dx} = - 3 e^{- 3x}$
$\Rightarrow\frac{d^2y}{dx^2}=9e^{-3x}$

$\text{Putting the values of }\frac{d^2 y}{d x^2}, \frac{dy}{dx}\text{ and y in (1), we get}$

$LHS = 9 e^{- 3x} - 3 e^{- 3x} - 6 e^{- 3x}$
$= 0$
$= RHS$

Thus, y = e−3x is the solution of the given differential equation.

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APPEARS IN

RD Sharma Class 12 Maths
Chapter 22 Differential Equations
Revision Exercise | Q 2 | Page 144