find-in-the-form-y-f-x-the-general-solution-of-the-differentail-equation-d-2-y-dx-2-dy-dx-6y-e-3x-

Question Number 87751 by Rio Michael last updated on 06/Apr/20

find in the form y = f (x) the general solution

of the differentail equation

\frac{d^{2} y}{{d x}^{2}} - \frac{d y}{d x} - 6 y = e^{3 x}

Commented by niroj last updated on 06/Apr/20

\frac{d^{2} y}{{dx}^{2}} - \frac{dy}{dx} - 6 y = e^{3 x}

let, \frac{d^{2} y}{{dx}^{2}} = D^{2} y and \frac{dy}{dx} = Dy

(D^{2} - D - 6) y = e^{3 x}

Auxilairy Equation,

m^{2} - m - 6 = 0

m^{2} - 3 m + 2 m - 6 = 0

m (m - 3) + 2 (m - 3) = 0

(m - 3) (m + 2) = 0

m = 3, - 2

Complementary Fuction = C_{1} e^{3 x} + C_{2} e^{- 2 x}

Particular Integral = \frac{e^{3 x}}{D^{2} - D - 6}

= \frac{x . e^{3 x}}{2 D - 1} = \frac{{xe}^{3 x}}{6 - 1} = \frac{{xe}^{3 x}}{5}

y = f (x) = CF + PI

y = C_{1} e^{3 x} + C_{2} e^{- 2 x} + \frac{{xe}^{3 x}}{5} / / .

Commented by Rio Michael last updated on 06/Apr/20

thanks sir