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seen Mar 19 at 23:41

Undergrad


Sep
26
answered Calculate spin wave function given probabilities of its alignment along 2 axes
Feb
7
comment Good examples of Ansätze
but of course this fails when the roots are degenerate and then we have to consider a totally separate ansatz, $y = x e^{kx}$, in order to generate the general solution.
Feb
7
comment Good examples of Ansätze
I don't think there is a general theorem that says that a single ansatz will generate the general solution to the problem... but if the problem is simple enough sometimes we know more properties about the system. for example, in the example you gave, $y'' + a y' + by = 0$, the dimension of the space of solutions is 2 (think Wronskian) - so if we get 2 linearly independently solutions we are done. Then it happens that the ansatz $e^{kx}$ usually gives 2 roots of $k$, telling us that our search for the general solution is over and it is given by $y = Ae^{k_1 x} + B e^{k_2 x}$.
Feb
3
comment How is this linear 2nd-order ODE solved?
ok.. tell me what the solution to this equation is: $\ddot{\phi} + \phi = 0$. then tell me what the solution to this equation is: $\ddot{\phi} + \phi = 1$.
Feb
3
comment How is this linear 2nd-order ODE solved?
E is a constant. Show your working.
Feb
3
comment How is this linear 2nd-order ODE solved?
er $f(\tau)$ is known. It is $-g_2 \phi_1^2 - \omega_1 \ddot{\phi_1}$ where you told me $\phi_1 = p_1 \cos(\tau + \alpha)$. If you're asking what to guess for the particular solution, I already gave it to you. Just plug it into the ODE and match terms to get $A B C D E$.
Feb
3
awarded  Teacher
Feb
1
answered How is this linear 2nd-order ODE solved?