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 Sep24 awarded Autobiographer Sep26 answered Calculate spin wave function given probabilities of its alignment along 2 axes Feb7 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. Feb7 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 $y = Ae^{k_1 x} + B e^{k_2 x}$. Feb3 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: $\ddot{\phi} + \phi = 1$. Feb3 comment How is this linear 2nd-order ODE solved? E is a constant. Show your working. Feb3 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 $A B C D E$. Feb3 awarded Teacher Feb1 answered How is this linear 2nd-order ODE solved?