Question

3d²y/dx²+dy/dx-14y=0y(0)=1y(0)=-1

Answer 1

$$3\frac{d²y}{dx²}+\frac{dy}{dx}-14y=0$$

The characteristic equation is as follows

$$3k^2+k-14=0$$

Roots

$$k_1=-7/3,\quad k_2=2$$

General solution of DE

$$y(x)=C_1e^{-7/3x}+C_2e^{2x}$$

Using initial condition, we obtain

$$y(0)=C_1+C_2=1\\ y'(0)=\frac{-7}{3}C_1+2C_2=-1$$

Roots:

$$C_1=9/13,\quad C_2=4/13$$

Finally, the particular solution of DE

$$y(x)=9/13e^{-7/3x}+4/13e^{2x}$$