Dear Wathek Chammam,

Assuming that the equation has a polynomial solution of degree n, the highest term in x on the left hand-side is n+1, and this must be 4 for the equation to hold, thus n=3. Then, equating the coefficients of x^4 on both sides of the equation, we see that the polynomial is monic, thus it has the form y(x)=x^3+bx^2+dx+e, with b,d,e real. Then,

differentiating, substituting into the equation, and equating the coefficients of the same degree terms on both sides, we end up to the following non-linear system of equations

2b+c=0, 2d+bc-3=0, 2e+dc-2b=0, and ec-d=2. Solving the first equation for c and substituting into the other three, we obtain the system

2d-2b^2-3=0, 2e-2db-2b=0, -2eb-d=2. Then, solving the second equation for e and substituting into the third, we obtain the equation 2(d+1)b^2=-2-d. If d equals -1, this equation is impossible, thus d is different from -1 and solving for b^2 we obtain

b^2=-(d+2)/2(d+1). Then, solving 2d-2b^2-3=0 for b^2 and comparing with the last equation yields -(d+2)/2(d+1)=(2d-3)/2 which is easily solved for d to give d=+-1/sqr(2). Then, 2d-3=+-sqr(2)-3

Hello,

you may also insert a formal series into the differential equations then prove that the relationship between its coefficients leads to an infinite sequence of non zero values.

Gabriel

thank you dear Spiros and Gabriel

In the same frame, is it easy to generalize the result for any positive polynomial P, ie

2 * y (x) (x + c) -y '(x) = P

BEST WISHES

Dear Wathek Chammam ,

In addition to the helpful answers of the followers,

I would like to add the following result:

The differential equation 2y(x+c)-dy/dx = 2 + 2x⁴

has a polynomial solutions if and only if c takes any one of the following complex values:

-(1/2)√2√(-√2-3)

(1/2)√2√(-√2-3)

-(1/2)√2√(√2-3)

(1/2)√2√(√2-3)

In fact the ODE :

2y(x+c)-dy/dx = 2 + 2x⁴

has the polynomial solution:

y = x³ - cx² + ((3+2c²)/2)x - (1/2)c(2c²+5)

where c is any of the mentioned complex values.

For clear symbols see the attached file.

Best regards

Colleagues, thank you for your answers. I am now looking for the next problem

In the same frame, is it easy to generalize the result for any positive polynomial P, ie

2 * y (x) (x + c) -y '(x) = P

c real number.

BEST WISHES