third order non linear equation differential equation y'y'''=y''

Posted on 2008-08-10 23:39 By chad (1)

Categories:

Math Advice (Student)

how do i go about finding all the polynomials of degree 5 or less that satisfy this equation?

i found the two obvisous families of the solution to be y=a and y=bx, where a and b are constants.....

no other

Comment on 2008-08-11 03:43 from jakubi ( Maple Rating 4

923)

polynomial solution of degree 5 or less:

d:=diff(y(x),x)*diff(y(x),x$3)-diff(y(x),x$2);
yp:=add(a||i*x^i,i=0..5);
eval(d,y(x)=yp):
collect(%,x):
{coeffs(%,x)};
solve(%,{a0,a1,a2,a3,a4,a5});
{a0 = a0, a1 = a1, a2 = 0, a3 = 0, a4 = 0, a5 = 0}

A first integral is y''=ln(C1*y').

Also

Comment on 2008-08-11 04:06 from alec ( Maple Rating 4

1497)

y=a+bx are the only polynomial solutions.

Look at it that way. If y is a polynomial of degree <3, then y'''=0, so the lhs is 0. So the r.h.s. should be also 0. That gives the solution y=a+bx.

If y is a polynomial of degree n>=3, the l.h.s is a polynomial of degree

(n-1)+(n-3) = 2n-4,

and the r.h.s. is a polynomial of degree n-2. That gives an equation

2n-4 = n-2,

from which n=2 that contradicts to n being >=3.

Alec

solve for y'y'''=y'

Comment on 2008-08-11 08:06 from wales05 (1)

solve for y'y'''=y'

Really?

Comment on 2008-08-11 19:02 from Robert Israel ( Maple Rating 4

1460)

If that's really what the equation is, it's trivial: the difference of the two sides of the equation factors as

y' (y''' - 1)

so there are two cases: y' = 0 and y''' = 1.

polysols

Comment on 2008-08-15 19:39 from dharr ( Maple Rating 2

158)

Maple has a command for this. It gives the single solution that Alec found above

d:=diff(y(x),x)*diff(y(x),x$3)-diff(y(x),x$2):

DEtools[polysols](d,y(x));

gives

[y(x) = _C3+_C4*x]