Solve the differential equation

[kcoe05]

Okay so I am stuck on two pretty similar problems. I will display both problems and show you where I am stuck at.

1) y"' - 5y' -2y = 0

I rewrote it as r^3 - 5y - 2 = 0

By inspection i see that -2 is a solution. But how do i find out if there are any other solutions? Would the easiest way be to use a calculator? I would rather not though.

2) 4y"' - 3y' + 1 = 0

rewrote as 4r^3 - 3r + 1 = 0

-1 a solution by inspection but once again having the same difficulty as the first one.

Thanks in advance for the help

 

[galactus]

Okay so I am stuck on two pretty similar problems. I will display both problems and show you where I am stuck at.

1) y"' - 5y' -2y = 0

I rewrote it as r^3 - 5y - 2 = 0

By inspection i see that -2 is a solution. But how do i find out if there are any other solutions? Would the easiest way be to use a calculator? I would rather not though.

Since we know that one solution is -2, then it factors into \(\displaystyle (r+2)(r^{2}-2r-1)=0\)

The solutions of the quadratic can be easily found from the quadratic formula. They are

\(\displaystyle r=\sqrt{2}+1, \;\ r=-\sqrt{2}+1\)

Therefore, the solution of said DE is \(\displaystyle y=C_{1}e^{-2t}+C_{2}e^{(-\sqrt{2}+1)t}+C_{3}e^{(\sqrt{2}+1)t}\)

The other can be done in an analagous manner.

 

[Deleted member 4993]

Okay so I am stuck on two pretty similar problems. I will display both problems and show you where I am stuck at.

1) y"' - 5y' -2y = 0

I rewrote it as r^3 - 5r - 2 = 0

(r + 2)(r[sup:3cnksk5x]2[/sup:3cnksk5x] - 2r - 1) = 0

(r + 2)[r - (1 + 2[sup:3cnksk5x]1/2[/sup:3cnksk5x])][r - (1 - 2[sup:3cnksk5x]1/2[/sup:3cnksk5x])] = 0

By inspection i see that -2 is a solution. But how do i find out if there are any other solutions? Would the easiest way be to use a calculator? I would rather not though.

2) 4y"' - 3y' + 1 = 0

rewrote as 4r^3 - 3r + 1 = 0

-1 a solution by inspection but once again having the same difficulty as the first one. <<< follow the procedure above
Thanks in advance for the help