Question

Find y as a function of x if
y(4)−10y′′′+25y′′=0,

y(0)=15, y′(0)=1, y′′(0)=25, y′′′(0)=0.

Answer 1

the y(4) means 4th derivative btw

Answer 2

4th degree linear?
So umm, the characteristic equation will be given by:\[\Large r^4-10r^3+25r^2\quad=\quad 0\]Factoring:\[\Large r^2\left(r^2-10r+25\right)\quad=\quad0\]Which further factors,\[\Large r^2(r-5)^2\quad=\quad 0\]

Answer 3

Hmmm so it looks like we have 2 sets of repeated roots.
\(\Large r=0, \qquad r=5\)

Answer 4

For repeated roots of this form, we multiply by x each time the root repeats if I remember correctly.

So our general solution would look something like:\[\Large y\quad=\quad c_1e^0+c_2xe^0+c_3e^5+c_4xe^5\]Which we could probably simplify a lil bit.\[\Large y\quad=\quad c_1+c_2x+(c_3+c_4x)e^5\]

Answer 5

And then we've got a bunch of initial data to deal with? Oh boy :b

Answer 6

\[ y^{(4)}−10y'''+25y''=0,\quad y(0)=15,\quad y'(0)=1,\quad y''(0)=25,\quad y'''(0)=0\]

\[ \mathcal L\Big\{y^{(4)}−10y'''+25y''\Big\}=\mathcal L\Big\{0\Big\}\]

Answer 7

\[\mathcal L\Big\{y^{(4)}\Big\}−10\mathcal L\Big\{y'''\Big\}+25\mathcal L\Big\{y''\Big\}=\mathcal L\Big\{0\Big\}\]
\[\Big(s^4Y-s^3y(0)-s^2y'(0)-sy''(0)-y'''(0)\Big)\\−10\Big(s^3Y-s^2y(0)-sy'(0)-y''(0)\Big)\\+25\Big(s^2Y-sy(0)-y'(0)\Big)=0\]