Question

In Problems 35–64 solve the given differential equation by undetermined coefficients.

$\mathbf{y}\text{'}\text{'}\mathbf{-}\mathbf{9}\mathbf{y}\mathbf{=}\mathbf{54}$

Short answer

$y=c_1{e}^{-3x}+c_2{e}^{3x}-6$

Step-by-step solution

Definition of homogeneous differential equation

A homogeneous differential equation is an equation containing a differentiation and a function, with a set of variables.

Find the complementary function

Consider the following differential equation:

$y\text{'}\text{'}-9y=54$

Consider$y=e^{mx}$as the solution of the differential equation,

Substitute,$y=e^{mx},y\text{'}=m{e}^{mx},y\text{'}\text{'}=m^2{e}^{mx}$into$y\text{'}\text{'}-9y=0$to obtain the auxiliary equation.

$$\begin{gathered} m^2-9=0 \\ {m}^2=9 \\ m=\pm 3 \\ {y}_h=c_1{e}^{-3x}+c_2{e}^{3x} \end{gathered}$$

The main aim is to solve the differential equation using the method of undetermined coefficients.

Find the particular integral

Use method of undetermined coefficients as follows.

Assume that the particular integral be$y_p=k$, since the non-homogeneous part is a constant.

This satisfies the given differential equation.

So, substitute$y_p=k$in the differential equation.

$$\begin{gathered} y\text{'}\text{'}-9y=54 \\ \left(k\right)\text{'}\text{'}-9\left(k\right)=54 \\ 0-9k=54 \\ k=-6 \end{gathered}$$

So, the particular integral is $y_p=-6$.

General solution of the differential equation

Now, the general solution to the given differential equation is,

$$\begin{gathered} y=y_c+y_p \\ =c_1{e}^{-3x}+c_2{e}^{3x}-6 \end{gathered}$$

Hence, the required differential equation is$y=c_1{e}^{-3x}+c_2{e}^{3x}-6$.