Question

Obtain a family of solutions for the homogeneous differential equation

Answer 1

a)

$$y''-8y'+12y=0$$

Auxiliary (characteristic) equation

$$r^2-8r+12=0$$

$$(r-2)(r-6)=0$$

$$r_1=2, r_2=6$$

The family of solutions of the given differemtial equation is

$$y=C_1e^{2t}+C_2e^{6t}$$

b)

$$y''+10y'+25y=0$$

Auxiliary (characteristic) equation

$$r^2+10r+25=0$$

$$(r+5)^2=0$$

$$r_1=r_2=-5$$

The family of solutions of the given differemtial equation is

$$y=C_1e^{-5t}+C_2te^{-5t}$$

c)

$$y''-8y'+65y=0$$

Auxiliary (characteristic) equation

$$r^2-8r+65=0$$

$$D=(-8)^2-4(1)(65)=-196<0$$

$$r_{1,2}=\dfrac{8\pm\sqrt{-196}}{2(1)}=4\pm7i$$

The family of solutions of the given differemtial equation is

$$y=C_1e^{4t}\cos(7t)+C_2e^{4t}\sin(7t)$$