Answer to Question #275860 in Differential Equations for JAY

Question

1. y'''+3y''+4y'-8y=0    ans, y= c₁e^x+e^-2x(c₂cos2x+c₃sin2x)
2.  y⁽⁴⁾-4y''=0                   ans, y=c₁+c₂x+c₃e^2x+c₄e^-2x

Answer 1

1.

Corresponding (auxiliary) equation

$$r^3+3r^2+4r-8=0$$

$$r^2(r-1)+4r(r-1)+8(r-1)=0$$

$$(r-1)(r^2+4r+8)=0$$

$$(r-1)((r+2)^2+4)=0$$

$$r_1=1, r_2=-2+2i, r_3=-2-2i$$

The general solution of the given differential equation is

$$y=c_1e^x+e^{-2x}(c_2\cos(2x)+c_3\sin(2x))$$

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

2.

Corresponding (auxiliary) equation

$$r^4-4r^2=0$$

$$r_1=r_2=0, r_3=2, r_4=-2$$

The general solution of the given differential equation is

$$y=c_1+c_2x+c_3e^{2x}+c_4e^{-2x}$$