(Solved): (d) A hyperbolic equilibrium point is the same as an unstable equilibrium point, so small changes ...

(d) A hyperbolic equilibrium point is the same as an unstable equilibrium point, so small changes in the initial condition near the point result in solutions that move away from the point as time increases. (2) True or False? Any vertical translation of a solution of an autonomous ODE \( \frac{d y}{d t}=f(y) \) is another somer of that same ODE. True or False? The phase line for the autonomous ODE \( \frac{d y}{d t}=f(y)=y^{3} \) has an equilibrium point at which is a source. True or False? An equilibrium point for a 2-dimensional system that is a center is stable. \( \left(=\right. \) Let \( T \) be the linear operator defied by \( T(y)=y^{\prime \prime}+2 y \), where \( y \) is a function of \( t \) (I also could have written \( T(f)=f^{\prime \prime}+2 f \), where \( f \) is a function of \( \left.t\right) \). Find the image of the function \( \sin (3 t) \) under \( T \). In other words, find \( T(\sin (3 t)) \). (a) \( 7 \cos (3 t) \) (b) \( -7 \sin (3 t) \) (c) \( 7 \sin (3 t) \) (d) \( -7 \cos (3 t) \) (f) Let \( u(t, x)=e^{-2 t} \cos (3 x) \). Which partial differential equation (heat equation) does \( u(t, x) \) solve? (a) \( \frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}} \), where \( k=2 \) (b) \( \frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}} \), where \( k=\frac{9}{2} \) (c) \( \frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}} \), where \( k=9 \) (d) \( \frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}} \), where \( k=\frac{2}{9} \)