# Thread: find the general solution

1. ## find the general solution

I am suppose to find a small integral root of the characteristic equation by inspection; then factor by division; and find the general solution of

y^(4) + y^(3) -3y''-5y'-2y=0

I am having problem with the first part of this trying to find the integral root, can someone please help me??

Thanks

2. Originally Posted by mandy123
I am suppose to find a small integral root of the characteristic equation by inspection; then factor by division; and find the general solution of

y^(4) + y^(3) -3y''-5y'-2y=0

I am having problem with the first part of this trying to find the integral root, can someone please help me??

Thanks
So our auxillary equation is

$\displaystyle m^4+m^3-3m^2-5m-2=0$

From the rational roots theorem we know the only possible rational roots are $\displaystyle \pm 1 \pm 2$

so we only have 4 things to check

$\displaystyle 2^4+2^3-3(2)^2-5(2)-2=0$ so m-2 is a factor

so by long or synthetic division we get

$\displaystyle (m-2)(m^3+3m^2+3m+1)$

we notice that this is the binomial expansion of $\displaystyle (m+1)^3$

so we end up with $\displaystyle (m-2)(m+1)^3=0$

so the solution is

$\displaystyle y=c_1e^{2x}+c_2e^{-x}+c_3xe^{-x}+c_4x^2e^{-x}$