# Help with third order, non-linear DE.

• Aug 9th 2008, 03:36 AM
dgmossman
Help with third order, non-linear DE.
consider the thrid order DE:

y'y'''=y''

Question: Find all polynomials y(x) of degree 5 orl ess that satisfy this equation(It turns out that these are all of the solutions that are polynomials).

How would I go about doing this using maple? Thanks in advance for any help.
• Aug 9th 2008, 07:00 AM
shawsend
Looks to me by inspection, the only polynomial that satisfies the equation is $P(x)=a+bx$.
• Aug 9th 2008, 11:59 AM
topsquark
I've done this in Mathematica before. It is more of an algorithm than a program, though. Let
$y(x) = ax^5 + bx^4 + cx^3 + dx + e$

Then calculate
$y'y''' - y''$
and set it equal to zero. Then you get a set of equations in a, b, c, d, and e by setting coefficients of powers of x equal to zero. In general you will get a system of equations to solve, which Maple can solve for you.

However, in this case inspection works a lot faster. ;)

-Dan
• Aug 9th 2008, 03:45 PM
shawsend
This is what I'd do in Mathematica: Define $P_5(x)$. Calculate the first, third, and second derivatives, form the right and left sides, equate coefficients and construct seven equations in six unknowns (y'y''' has x^6 term), then use Solve to find the solution. Note that Solve reports c, d, e, and f are zero. Not sure why it's posting the results multiple times. Hopefully, you can translate it to Maple if you wish.

Code:

```In[25]:= val = a + b*x + c*x^2 + d*x^3 + e*x^4 + f*x^5; d1 = D[val, {x, 1}]; d3 = D[val, {x, 3}]; d2 = D[val, {x, 2}]; leftside = d1*d3; rightside = d2; eqns = Table[   Coefficient[leftside, x, n] == Coefficient[rightside, x, n], {n, 0,   6}] Solve[eqns, {a, b, c, d, e, f}] Out[31]= {6 b d == 2 c, 12 c d + 24 b e == 6 d,  18 d^2 + 48 c e + 60 b f == 12 e, 96 d e + 120 c f == 20 f,  96 e^2 + 210 d f == 0, 360 e f == 0, 300 f^2 == 0} During evaluation of In[25]:= Solve::svars: Equations may not give \ solutions for all "solve" variables. >> Out[32]= {{c -> 0, f -> 0, e -> 0, d -> 0}, {c -> 0, f -> 0, e -> 0,   d -> 0}, {c -> 0, f -> 0, e -> 0, d -> 0}, {c -> 0, f -> 0, e -> 0,   d -> 0}}```
 results in 7 equations in 6 unknowns (y'y''' has x^6 term)