Question

Use the procedures developed to find the general solution of the differential equation. (Let x be the independent variable.)

2y''' + 15y'' + 24y' + 11y= 0

Answer 1

Solution :

Given :

2y''' + 15y'' + 24y' + 11y= 0

Let x = independent variable

$(a_0D^n + a_1D^{n-1}+a_2D^{n-2} + ....+ a_n) y) = Q(x)$  is a differential equation.

If $Q(x) \neq 0$

It is non homogeneous then,

The general solution  = complementary solution + particular integral

If Q(x) = 0

It is called the homogeneous then the general solution =  complementary solution.

2y''' + 15y'' + 24y' + 11y= 0

$(2D^3+15D^2+24D+11)y=0$

Auxiliary equation,

$2m^3+15m^2+24m +11 = 0$

-1  | 2    15    24     11

    | 0   -2    - 13    -11  

      2    13    11       0

∴ $2m^2+13m+11=0$

The roots are

$=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

$=\frac{-13\pm \sqrt{13^2-4(11)(2)}}{2(2)}$

$=\frac{-13\pm9}{4}$

$=-5.5, -1$

So, $m_1, m_2, m_3 = -1, -1, -5.5$

Then the general solution is :

$= (c_1+c_2 x)e^{-x} + c_3 \ e^{-5.5x}$