# What is the general solution of the differential equation?  z'''-5z''+25z'-125z=1000

Dec 31, 2017

$z \left(x\right) = {e}^{5 x} + A \cos \left(5 x\right) + B \sin \left(5 x\right) - 8$

#### Explanation:

Assuming that we have:

$z ' ' ' - 5 z ' ' + 25 z ' - 125 z = 1000$ .... [A]

This is a third order linear non-Homogeneous Differentiation Equation. The standard approach is to find a solution, ${z}_{c}$ of the homogeneous equation by looking at the Auxiliary Equation, which is the quadratic equation with the coefficients of the derivatives, and then finding an independent particular solution, ${z}_{p}$ of the non-homogeneous equation.

Complementary Function

The homogeneous equation associated with [A] is

$z ' ' ' - 5 z ' ' + 25 z ' - 125 z = 0$

And it's associated Auxiliary equation is:

${m}^{3} - 5 {m}^{2} + 25 m - 125 = 0$

The hardest part with higher order DE is solving this equation. If we consider the graph $y = {x}^{3} - 5 {x}^{2} + 25 x - 125$:
graph{y = x^3-5x^2+25x-125 [-10, 10, -30 30]}

We note there is one real solution at $x = 5$, with this in mind we can factorise the cubic auxiliary equation:

$\left(m - 5\right) \left({m}^{2} + 25\right)$

And so we have one real real $m = 5$ and two pure imaginary roots $m = \pm 5 i$

Thus the solution of the homogeneous equation is:

${z}_{c} = A {e}^{5 x} + B \cos \left(5 x\right) + C \sin \left(5 x\right)$

Particular Solution

With this particular equation [A], a probable solution is of the form:

$z = a$

Where $a$ is a constant coefficient to be determined. Let us assume the above solution works, in which case be differentiating wrt $x$ we have:

$z ' = z ' ' = z ' ' ' = 0$

Substituting into the initial Differential Equation $\left[A\right]$ we get:

$= 0 - 0 + 0 - 125 a = 1000 \implies a = - 8$

And so we form the Particular solution:

${z}_{p} = - 8$

General Solution

Which then leads to the GS of [A}

$z \left(x\right) = {z}_{c} + {z}_{p}$
$\setminus \setminus \setminus \setminus \setminus \setminus \setminus = A {e}^{5 x} + B \cos \left(5 x\right) + C \sin \left(5 x\right) - 8$