# Linear System of Equations

#### jameselmore91

I'm having some trouble with Linear Systems of Equations, I get the basic concept but it would be helpful if someone could do the following problem step by step.

$$\displaystyle u'\; =\; 4u\; -\; v$$
$$\displaystyle v'\; =\; -4u\; +\; 4v$$

Where v and u are dependent over x.

MHF Helper

#### HallsofIvy

MHF Helper
I'm having some trouble with Linear Systems of Equations, I get the basic concept but it would be helpful if someone could do the following problem step by step.

$$\displaystyle u'\; =\; 4u\; -\; v$$
$$\displaystyle v'\; =\; -4u\; +\; 4v$$

Where v and u are dependent over x.
The most fundamental way to do this is to differentiate the first equation again gettin u''= 4u'- v'. From the second equation, v'= -4u+ 4v so we can replace it: u''= 4u'- (-4u+ 4v)= 4u'+ 4u- 4v. We still have a "v" to eliiminate but from the first equation, v= 4u- u' so that becomes u''= 4u'+ 4u- 4(4u- u')= 8u' -12u. that is, u''- 8u+ 12u= 0. Solve that equation for the general solution, u, then use v= 4u- u' to find v.

dwsmith