# Thread: Help with third order, non-linear DE.

1. ## 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.

2. Looks to me by inspection, the only polynomial that satisfies the equation is $\displaystyle P(x)=a+bx$.

3. I've done this in Mathematica before. It is more of an algorithm than a program, though. Let
$\displaystyle y(x) = ax^5 + bx^4 + cx^3 + dx + e$

Then calculate
$\displaystyle y'y''' - y''$
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

4. This is what I'd do in Mathematica: Define $\displaystyle P_5(x)$. 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)