Question

(d^(2)y)/(dx^(2)) 5(dy)/(dx) 6x
solve​

Answer 1

Answer:

$y = a{e}^{2x} + b {e}^{3x}$

Step-by-step explanation:

To solve,

$\dfrac{ {d}^{2}y }{d {x}^{2} } - 5 \dfrac{dy}{dx} + 6y = 0$

Let,

$y = a {e}^{kx}$

and so

$\dfrac{dy}{dx} = \dfrac{d(a {e}^{kx} )}{dx} = ak {e}^{kx}$

then,

$\dfrac{d^{2} y}{d {x}^{2} } = \dfrac{d(ak {e}^{kx} )}{dx} = a {k}^{2} {e}^{kx}$

Now substituting these in the equation:

$a {k}^{2} {e}^{kx} - 5ak {e}^{kx} + 6a {e}^{kx} = 0 \\ \implies {k}^{2} - 5k + 6 = 0 \\ \implies \: {k}^{2} - 3k - 2k + 6 = 0 \\ \implies k(k - 3) - 2(k - 3) = 0 \\ \implies \: (k - 3)(k - 2) = 0$

Therefore k = 2,3

Since the roots are different therefore,

The solution will be,

$y = a{e}^{2x} + b {e}^{3x}$