Question

1.  y⁽⁴⁾+8y''+16y=0       ans, y=(c₁+c₂x)cos2x+(c₃+c₄x)sin2x
2. y⁽⁴⁾+4y''+4y=0          ans, y=(c₁+c₂x)cos $sqr2$ x+(c₃+c₄x)sin$sqr2$ x

Answer 1

1.

Corresponding (auxiliary) equation

$$r^4+8r^2+16=0$$

$$(r^2+4)^2=0$$

$$r^2=-4$$

$$r=\pm 2i$$

The general solution of the given differential equation is

$$y=c_1\cos(2x)+c_2x\cos(2x)$$

$$+c_3\sin(2x)+c_4x\sin(2x)$$

2.

Corresponding (auxiliary) equation

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

$$(r^2+2)^2=0$$

$$r^2=-2$$

$$r=\pm i\sqrt{2}$$

The general solution of the given differential equation is

$$y=c_1\cos(\sqrt{2}x)+c_2x\cos(\sqrt{2}x)$$

$$+c_3\sin(\sqrt{2}x)+c_4x\sin(\sqrt{2}x)$$