# Linear homogeneous differential equations of 2nd order Step-By-Step

$$11 y{\left(x \right)} - 2 \frac{d}{d x} y{\left(x \right)} + 3 \frac{d^{2}}{d x^{2}} y{\left(x \right)} = 0$$

### Detail solution

Divide both sides of the equation by the multiplier of the derivative of y'':$$3$$

We get the equation:

$$\frac{11 y{\left(x \right)}}{3} - \frac{2 \frac{d}{d x} y{\left(x \right)}}{3} + \frac{d^{2}}{d x^{2}} y{\left(x \right)} = 0$$

This differential equation has the form:

y'' + p*y' + q*y = 0,

where

$$p = - \frac{2}{3}$$

$$q = \frac{11}{3}$$

It is called

**linear homogeneous**

second-order differential equation with constant coefficients.

The equation has an easy solution

We solve the corresponding homogeneous linear equation

y'' + p*y' + q*y = 0

First of all we should find the roots of the characteristic equation

$$q + \left(k^{2} + k p\right) = 0$$

In this case, the characteristic equation will be:

$$k^{2} - \frac{2 k}{3} + \frac{11}{3} = 0$$

- this is a simple quadratic equation

The roots of this equation:

$$k_{1} = \frac{1}{3} - \frac{4 \sqrt{2} i}{3}$$

$$k_{2} = \frac{1}{3} + \frac{4 \sqrt{2} i}{3}$$

As there are two roots of the characteristic equation,

solving the correspondent differential equation looks as follows:

$$y{\left(x \right)} = e^{k_{1} x} C_{1} + e^{k_{2} x} C_{2}$$

The final answer:

$$y{\left(x \right)} = C_{1} e^{x \left(\frac{1}{3} - \frac{4 \sqrt{2} i}{3}\right)} + C_{2} e^{x \left(\frac{1}{3} + \frac{4 \sqrt{2} i}{3}\right)}$$