I'm attepmting to solve a second order partial differential equation using the seperation of variable method, but have gotten up to where i have reduced to ordinary differential equations and has been so long since i did them i'm confused.
If i have say;
X'' - X/(k^2) = 0
And i try a solution of the form X =
hence X' =
and X'' =
when substituted in gives;
is this correct up to here? and the following is where i'm confused,
this seems to me to be real distinct solutions hence general solution should be;
but i remember the lecturer saying that this should be a trig or hyperbolic solution, as there is no complex roots i'm gussing hyperbolics?
Any help would be greatly appreciated, i'm very hazy on this stuff,
yes k is a real number such that k > 0,
in my notes there is an example where;
X'' + (k^2) X = 0
When using trial solution
so general solution is;
which goes to;
X = Acos(kx) + Bsin(kx)
which i don't understand as;
1/2( (e^ix) + (e^-ix) ) = cos(x) so how is the solution a combination of cos and sin ?
Yes, and you should also know that .
Multiplying the second equation by i, and adding that to the first, , the formula Prove It gave. Subtracting that from the first gives .
From that, with A= a+ b and B= a- b. If a and b were equal, that would be only a "cos(x)" term. It is because they are not equal that you have both cos(x) and sin(x).