A Treatise On the Calculus of Finite Differences
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n.).
AKT. 7.] SIMULTANEOUS EQUATIONS.
229 7. The solution of two very remarkable non-linear equa- tions has been deduced by Prof. Sylvester from that of linear equations with constant coefficients.
Let u * +? +Pi M * + »- l + & c. +p n u x = (36) be any such equation. Then writing it down for the next n values of x *Wn +j>i«*4.» + &o- +P«u x+1 = 0, &c. = 0, Eliminating the quantities p v p s , &c. we obtain .u„ u u ,* .... u ,» = 0.
.(37), an equation which must be satisfied by every solution o...f (36).
Now the solution of (36) is u x = Acf + Bp' + &c. ton. terms (38), where A, B, ... are arbitrary and a, j3, ... depend on p v p 2 , ... and these last do not appear in equation (37) which we are now considering. Hence (38) will be the solution of (37), a, /3, ... being also considered arbitrary, thus making the full number of 2m arbitrary constants.
By a slight variation in the method of elimination we can obtain the solution of a yet more general equation. Taking the last term of each of the equations to the other side and eliminating^, p v ...p„_ v we obtain U.
What to read after A Treatise On the Calculus of Finite Differences?
You can find similar books in the "Read Also" column, or choose other free books by Boole, George, 1815-1864 to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
n.).
AKT. 7.] SIMULTANEOUS EQUATIONS.
229 7. The solution of two very remarkable non-linear equa- tions has been deduced by Prof. Sylvester from that of linear equations with constant coefficients.
Let u * +? +Pi M * + »- l + & c. +p n u x = (36) be any such equation. Then writing it down for the next n values of x *Wn +j>i«*4.» + &o- +P«u x+1 = 0, &c. = 0, Eliminating the quantities p v p s , &c. we obtain .u„ u u ,* .... u ,» = 0.
.(37), an equation which must be satisfied by every solution o...f (36).
Now the solution of (36) is u x = Acf + Bp' + &c. ton. terms (38), where A, B, ... are arbitrary and a, j3, ... depend on p v p 2 , ... and these last do not appear in equation (37) which we are now considering. Hence (38) will be the solution of (37), a, /3, ... being also considered arbitrary, thus making the full number of 2m arbitrary constants.
By a slight variation in the method of elimination we can obtain the solution of a yet more general equation. Taking the last term of each of the equations to the other side and eliminating^, p v ...p„_ v we obtain U.
What to read after A Treatise On the Calculus of Finite Differences?
You can find similar books in the "Read Also" column, or choose other free books by Boole, George, 1815-1864 to read onlineMoreLess
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