 A brine tank of volume \(V\) get an influx of \(p\%\) brine at the rate \(r\), which mixes instantaneously and flows away at the same rate \(r\). If the tank was filled with fresh water initially, when it will have
Archive  courses
RSS feed for this sectionseptember 26
old (mock) midterm
 Consider the force function
\[
f(y)=\sin(y)+1/2.
\]
Draw the graph of the potential function
\[
U(y)=\int_0^y f(\eta)d\eta.
\]  Sketch the level sets of \( v^2/2+U(y)=C\) for \(C=1,0,1\) on \((y,v)\) plane.
 Sketch the solutions of
\[
y”=\sin(y)+1/2,
\]
passing through \((0,0)\).

Draw the graph of the potential function
september 25
\(\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}\)
Lyapunov’s direct method, cont’d
Convenient Lyapunov functions: homogeneous ones.
Strict homogeneous Lyapunov function for a linear system remains one after a perturbation of the system.
Examples.
Good homogeneous functions: quadratic forms.
Quadratic forms
Coordinate forms. …
math285: the week of september 22
There will be no new homework: review the material covered thus far for the upcoming midterm (on 9.29). The midterm problems will be similar to those addressed in homework and in the class.…
september 19
 Sketch the vector fields for each value of the parameter; indicate equilibria and their stability:
 \[
x’=3\sin(x)+x+a, a=3,0,3;
\]  \[
x’=\tan(x)3x+a, a=5,0,5;
\]  \[
x’=e^xx+a, a=1,0,1;
\]
 \[
 Find the three consecutive Picard iterations for the equation.
Solve the differential equation;
ece515: homework 2
\(\def\Real{\mathbb{R}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}
\)
Due by midnight of september 25.
 Consider the space \(V\) of polynomials (with real coefficients) of degree \(\leq 4\), and the operator \(D:V\to V\) of differentiation. Compute
\[
\exp(tD).
\]  Let
september 18
\(\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}\)
Remarks
 What operators can appear as state transition (fundamental) ones, \(\Phi(t)\)?
For LTVs, any operator \(\Phi\) with \(\det(\Phi)>0\) can be a fundamental solution; for LTIs the situation is far more complicated (say, the
september 16
\(\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}\)
linear time dependent systems
Generalizing the notion of state transition matrix, we can address also the timedependent systems
\[
\dot{x}(t)=A(t)x(t).
\]
Here, we dove the equation on \([t_1,t_2]\) with \(x(t_1)=x\); the value of solution at …
math285: the week of september 15
september 12
 Solve linear ODE of 1st order:
 \[
y’+y\tan{x}=\sec{x};
\]  \[
xy+e^x=xy’;
\]  \[
(xy’1)\ln{x}=2y.
\]
 \[
 Reduce the ODE to a linear one, and solve:
 \[
(x+1)(y’+y^2)=y;
\]  \[
xy^2y’=^2+y^3;
\]  \[
xyy’=y^2+x.
\]
 \[
 Reduce order of the equation, and solve: