Question

Q: find a general solution of the differential equation. How to solve it? Thank you! (pictures below)

here:

result:

Answer 1

See below

Explanation:

TO SOLVE:

• `Delta^2 (a_n) = Delta(5n+1 +a_n)`

Definitions:

• `Delta (a_n)=a_(n+1)-a_n`

• `Delta^2(a_n)= Delta (a_(n+1))-\Delta (a_n) = a_(n+2)-2 a_(n+1)+ a_n`

`Delta^2 (a_n) = 5 + Delta (a_n)`

`Delta^2 (a_n) - Delta a_n = 5`

The final equation is:

`color(blue)( a_(n+2) - 3 a_(n+1) + 2 a_n = 5 ) qquad square`

Assume:

• `a_(n+1) = lambda a_n, qquad lambda = " const"`

• `implies a_(n+2) = lambda a_(n+ 1) = lambda^2 a_n " etc"`

The null solution for `square` is:

`lambda^2 a_n - 3 lambda a_n + 2 a_n = 0`

`implies lambda = 1,2`

Solutions being:

• `a_(n+1) = a_n implies a_n = " const" = C_1`

• `a_(n+1) = 2 a_n implies a_n = C_2 * 2^n`

For the particular solution, assume:

• `a_n = alpha n + beta`

• `implies a_(n+1) = alpha (n+1) + beta, " etc"`

`implies alpha (n+2) + beta - 3 (alpha (n+1) + beta) + 2 (alpha n + beta) = 5`

`implies alpha = - 5`, with `beta` as a free variable

Superposition of all solutions:

• `a_n = C_1 + C_2 * 2^n - 5n` [ie `beta` included within `C_1`]