# [SOLVED]17.2.01 Solve the given equation by the method of undetermined coefficients

#### karush

##### Well-known member
$\tiny{17.2.01}\\$
$\textrm{ Solve the given equation by the method of undetermined coefficients.}$
\begin{align*}\displaystyle
y''+7y'+10y&=80
\end{align*}
$\textit{this is in the form}$
\begin{align*}\displaystyle
x^2+7x+10&=0\\
(x+2)(x+5)&=0
\end{align*}
$\textit{this is a second-order linear ordinary differential equation so}$
\begin{align*}\displaystyle
y& = c_1 e^{-5 x} + c_2 e^{-2 x}+8
\end{align*}

ok I know stuff is missing here???

#### MarkFL

Staff member
You look at what's on the RHS and see it is a constant, and you also note that no term of the homogeneous solution is a constant, therefore you assume the particular solution must be a constant:

$$\displaystyle y_p(x)=A$$

And so:

$$\displaystyle y_p'(x)=0$$

$$\displaystyle y_p''(x)=0$$

Substituting $y_p$ into the ODE, we obtain:

$$\displaystyle 0+0+10A=80\implies A=8$$

Add so we have:

$$\displaystyle y_p(x)=8$$

Now, using the principle of superposition, we find the solution to the ODE to be:

$$\displaystyle y(x)=y_h(x)+y_p(x)=c_1e^{-5x}+c_2e^{-2x}+8$$

#### karush

##### Well-known member
think I got it
let me try another one.

my class here is over so it will the Summer of Review..

and a peek thru the door of Calc III

of which I have heard horror stories