Finding unknown coefficients of a polynomial

Hi,

Matlab Version 2010b

I have a 7 degree polynomial function with unknown coefficients x=a0+a1*y+a2*y^2+a3*y^3+a4*y^4+a5*y^5+a6*y^6+a7*y^7

further y is a function of t : y=t^3

D_x means first derivative of x with respect to time (t)

DD_x means second derivative of x with respect to time (t)

I have the following values:

at t=t1

D_x=0 .... (1) DD_x=0 .... (2)

at t=t2

DD_x=0 ....(3)

at t=t3

DD_x=0 .....(4)

at t=t4

D_x+54*DD_x=0 ...(5)

at t=t5

x+D_x*10+DD_x*54=12 ...(6)

at t=t6

x + 87*DD_x=21 ....(7) D_x +89*DD_x=0 ...(8)

so we have 8 equations and 8 unknowns (the unknown coefficients)

How do i find the values of a0 a1 a2 a3 a4 a5 a6 a7 in terms of t1 t2 t3 t4 t5 t6?

Please help me out. I hope I am clear

Answer 1

[John D'Errico]

Pointing you in a reasonable direction...

syms t y a0 a1 a2 a3 a4 a5 a6 a7
syms t1 t2 t3 t4 t5 t6
x=a0+a1*y+a2*y^2+a3*y^3+a4*y^4+a5*y^5+a6*y^6+a7*y^7;
x = subs(x,y,t^3)
x =
a7*t^21 + a6*t^18 + a5*t^15 + a4*t^12 + a3*t^9 + a2*t^6 + a1*t^3 + a0

>> x1 = subs(diff(x,t),t,t1)
x1 =
21*a7*t1^20 + 18*a6*t1^17 + 15*a5*t1^14 + 12*a4*t1^11 + 9*a3*t1^8 + 6*a2*t1^5 + 3*a1*t1^2

See that x1 is simply a linear function of the a coefficients.

You can write each of your other equations as simple variations of the above. Then use solve.

John