Question

Solve the following ordinary differential equation.

=> d^2y/dx^2 + dy/dx + y = 0 .

Answer 1

Solution. The characteristic equation for a ordinary differential equation

$$k^2+k+1=0$$

Solve the quadratic equation

$$D=1^2-4\times 1 \times1=-3<0$$

Roots of the equation

$$k_1=\frac{-1-\sqrt{3}i}{2}$$

$$k_2=\frac{-1+\sqrt{3}i}{2}$$

Therefore the solution for a ordinary differential equation

$$y=e^{\frac{-x}{2}}(C_1sin(\frac{\sqrt{3}x}{2})+C_2cos(\frac{\sqrt{3}x}{2}))$$

where C1 and C2 are constants.

Answer.

$$y=e^{\frac{-x}{2}}(C_1sin(\frac{\sqrt{3}x}{2})+C_2cos(\frac{\sqrt{3}x}{2}))$$

where C1 and C2 are constants.