Toronto Math Forum

Hi,

I'm confused as to exactly how we can determine the nature of a critical point in a nonlinear system if the eigenvalues of the corresponding linear system are real and repeated or purely imaginary. I understand that in this case the usual theorem (9.3.3) does not apply, but I do not know how to determine if the nonlinear terms affect the system significantly or not.

Thanks!

Hi,
In the textbook, it says that a system of first-order equations can sometimes be transformed into a single higher-order equation (by the process given in problem 7 on the 7.1 problems).
Am I correct in saying that this is only possible to do in general when the determinant of the coefficient matrix associated with the system is nonzero?
I.e. if $\left\{\begin{aligned}
&x'_1= ax_1 + bx_2\\
&x'_2= cx_1 + dx_2
\end{aligned}\right.$, this can be converted into a singular equation iff $ad - bc$ $\neq 0$?
Thank you!   

Hi, I have two questions respectively about 3.3 problem 34 and 35.
1) In problem 34, we are asked to make a change of variables from $t$ to $x=\ln{t}$. This is fine - I substituted in my equations for $\frac{d^2y}{dt^2}$ and $\frac{dy}{dx}$ and got the desired result. However, I am not exactly clear on what happens to the term $by$, which doesn't change even though the variable $y$ is represented in technically does. Is it okay to leave it like that since we essentially end up back substituting when writing the roots of the characteristic equation?
2) In problem 35, finding the roots of the characteristic equation yields $r_1 = i$ and $r_2 = -i$. Is it sufficient to write the equation $y(t) = C_1t^i + C_2t^{-i}$, or should I use the same justification as is given in section 3.3 to write it as $y(t) = C_1\cos{ln(t)} + C_2\sin{ln(t)}$ so that we have real solutions?

Hello,
I have noticed that in the textbook, initial value problems are given as $y(t_0) = y_0$ and $y'(t_0) = y'_0$. That is, both initial conditions are defined at $t_0$ and moreover, the related Wronskian is also always evaluated at $t_0$. What happens if the initial conditions are defined at different $t_0, t_1$? Moreover, what if $t_0$ and $t_1$ are in different rectangles (meaning the areas where the functions $p, q, g$ are continuous)?
Thanks