# What is the solution to the Differential Equation (d^2y)/(dx^2) - 6dy/dx = 54x + 18?

Jul 21, 2017

$y \left(x\right) = A + B {e}^{6 x} - \frac{9}{2} {x}^{2} - \frac{9}{2} x$

#### Explanation:

We have:

$\frac{{d}^{2} y}{{\mathrm{dx}}^{2}} - 6 \frac{\mathrm{dy}}{\mathrm{dx}} = 54 x + 18$

This is a second order linear non-Homogeneous Differentiation Equation. The standard approach is to find a solution, ${y}_{c}$ 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, ${y}_{p}$ of the non-homogeneous equation.

Complementary Function

The homogeneous equation associated with [A] is

$y ' ' - 6 y ' + 0 y = 0$

And it's associated Auxiliary equation is:

${m}^{2} - 6 m + 0 = 0$
${m}^{2} - 6 m = 0$
$m \left(m - 6\right) = 0$

Which has two real and distinct solutions $m 0 , 6$

Thus the solution of the homogeneous equation is:

${y}_{c} = A {e}^{0 x} + B {e}^{6 x}$
$\setminus \setminus \setminus = A + B {e}^{6 x}$

Particular Solution

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

$y = a {x}^{2} + b x + c$

Where $a , b , c$ are constants to be determined by substitution

Let us assume the above solution works, in which case be differentiating wrt $x$ we have:

$y ' \setminus \setminus = 2 a x + b$
$y ' ' = 2 a$

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

$2 a - 6 \left(2 a x + b\right) = 54 x + 18$
$\therefore 2 a - 12 a x - 6 b = 54 x + 18$

Equating coefficients of ${x}^{0}$ and $x$ we get:

${x}^{0} : 2 a - 6 b = 18$
${x}^{1} : - 12 a = 54$

Solving simultaneous we have:

$a = - \frac{9}{2} , b = - \frac{9}{2}$

And so we form the Particular solution:

${y}_{p} = - \frac{9}{2} {x}^{2} - \frac{9}{2} x$

General Solution

Which then leads to the GS of [A}

$y \left(x\right) = {y}_{c} + {y}_{p}$
$\setminus \setminus \setminus \setminus \setminus \setminus \setminus = A + B {e}^{6 x} - \frac{9}{2} {x}^{2} - \frac{9}{2} x$

As we have a linear combination of three linearly independent solutions, this is the GS.