Question

1.  y'' + 4y' + 7y = 0         ans, y= e^-2x(c₁cos$sqr3$ x+c₂sin$sqr3$ x)

Answer 1

1.

Corresponding (auxiliary) equation

$$r^2+4y+7=0$$

$$D=(4)^2-4(1)(7)=-12$$

$$r=\dfrac{-4\pm \sqrt{12}}{2}=-2\pm i\sqrt{3}$$

The general solution of the given differential equation is

$$y=c_1e^{-2x}\cos(\sqrt{3}x)+c_2e^{-2x}\sin(\sqrt{3}x)$$