differential equations with constant coefficients

**sinichko** · May 29th 2010, 12:21 PM

Please help me with this one:
$y''-9y'+20y=x^{2}e^{4x}$

**galactus** · May 29th 2010, 01:01 PM

Lets use Variation of Parameters. I have always preferred that method, but that's me.

From the auxiliary equation we get:

$m^{2}-9m+20=(m-5)(m-4), \;\ m=5, \;\ m=4$

$y_{c}=C_{1}e^{4x}+C_{2}e^{5x}$

Now use the determinants to find W, W1, W2:

$W=\begin{vmatrix}y_{1}&y_{2}\\y_{1}^{'}&y_{2}^{'}\ end{vmatrix}=\begin{vmatrix}e^{4x}&e^{5x}\\4e^{4x} &5e^{5x}\end{vmatrix}=e^{9x}$

$W_{1}=\begin{vmatrix}0&y_{2}\\f(x)&y_{2}^{'}\end{v matrix}=\begin{vmatrix}0&e^{5x}\\x^{2}e^{4x}&5e^{5 x}\end{vmatrix}=-x^{2}e^{9x}$

$W_{2}=\begin{vmatrix}y_{1}&0\\y_{1}^{'}&f(x)\end{v matrix}=\begin{vmatrix}e^{4x}&0\\4e^{4x}&x^{2}e^{4 x}\end{vmatrix}=x^{2}e^{8x}$

$u_{1}^{'}=\frac{W_{1}}{W}=-x^{2}$

$u_{2}^{'}=\frac{W_{2}}{W}=\frac{x^{2}}{e^{x}}$

Integrate to find u1 and u2:

$u_{1}=-\int x^{2}dx=\frac{-1}{3}x^{3}$

$u_{2}=\int\frac{x^{2}}{e^{x}}dx=\frac{-x^{2}-2x-2}{e^{x}}$

$y_{p}=u_{1}+u_{2}$

$\boxed{y=y_{c}+y_{p}=\left(\frac{-1}{3}x^{3}-x^{2}-2x-2\right)e^{4x}+C_{1}e^{4x}+C_{2}e^{5x}}$

There is a nice outline for future reference.

**sinichko** · May 29th 2010, 10:07 PM

Thanks a lot!

Thread: differential equations with constant coefficients

LinkBack

Thread Tools

Display

differential equations with constant coefficients

Similar Math Help Forum Discussions

System of Linear Differential Equations with Constant Coefficients

Second order linear homogenous Differential Equation with constant coefficients

2nd order linear Differential Equations with constant coefficients.

second order differential constant coefficients homogeneous

Linear Equations of Higher Order: Homogeneous Equations With Constant Coefficients

Search Tags