Question

1. y'' + 4y' + 7y = 0      ans, y= e^-2x(c₁cosx+c₂sinx)
2. 2y'' - 7y' + 3y = 0     ans, y=c₁e^3x+c₂e^1/2x

Answer 1

1.

Corresponding (auxiliary) equation

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

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

$$r=\dfrac{-4\pm\sqrt{-12}}{2(1)}=-2\pm i\sqrt{3}$$

The general solution of the given differential equation is

$$y=e^{-2x}(c_1\cos(\sqrt{3}x)+c_2\sin(\sqrt{3}x))$$

2.

Corresponding (auxiliary) equation

$$2r^2-7r+3=0$$

$$D=(-7)^2-4(2)(3)=25$$

$$r=\dfrac{7\pm\sqrt{25}}{2(2)}=\dfrac{7\pm5}{4}$$

$$r_1=\dfrac{7+5}{4}=3, r_2=\dfrac{7-5}{4}=\dfrac{1}{2}$$

The general solution of the given differential equation is

$$y=c_1e^{3x}+c_2e^{x/2}$$