Question

d^2y/dx^2+dy/dx-y=6

Answer 1

$z=y+6, \dfrac{dy}{dx}=\dfrac{dz}{dx}, \dfrac{d^2z}{dx^2}=\dfrac{d^2y}{dx^2}$

The homogeneous differential equation

$$\dfrac{d^2z}{dx^2}+\dfrac{dz}{dx}-z=0$$

The corresponding (auxiliary) equation

$$r^2+r-1=0$$

$$D=(1)^2-4(1)(-1)=5$$

$$r=\dfrac{-1\pm\sqrt{5}}{2(1)}$$

The general solution of the homogeneous differential equation is

$$z=C_1e^{({-1-\sqrt{5} \over 2})x}+C_2e^{({-1+\sqrt{5} \over 2})x}$$

$$y=C_1e^{({-1-\sqrt{5} \over 2})x}+C_2e^{({-1+\sqrt{5} \over 2})x}-6$$