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

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