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## september 26

• 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

## old (mock) midterm

1. Consider the force function
$f(y)=\sin(y)+1/2.$
1. Draw the graph of the potential function
$U(y)=-\int_0^y f(\eta)d\eta.$
2. Sketch the level sets of $$v^2/2+U(y)=C$$ for $$C=-1,0,1$$ on $$(y,v)$$ plane.
3. Sketch the solutions of
$y”=\sin(y)+1/2,$
passing through $$(0,0)$$.

## 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

1. Sketch the vector fields for each value of the parameter; indicate equilibria and their stability:
• $x’=3\sin(x)+x+a, a=-3,0,3;$
• $x’=\tan(x)-3x+a, a=-5,0,5;$
• $x’=e^x-x+a, a=-1,0,1;$
2. 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.

1. Consider the space $$V$$ of polynomials (with real coefficients) of degree $$\leq 4$$, and the operator $$D:V\to V$$ of differentiation. Compute
$\exp(tD).$
2. 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 time-dependent 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

Review sections 2.2, 2.4, 2.6 of the textbook (Edwards and Penney). Listen to the video lectures.

Additional reading (highly recommended!): chapter 1 of Ordinary differential equations : a practical guide by
Bernd J. Schroers (CUP, 2011).

Homework (due by …

## september 12

1. Solve linear ODE of 1st order:
1. $y’+y\tan{x}=\sec{x};$
2. $xy+e^x=xy’;$
3. $(xy’-1)\ln{x}=2y.$
2. Reduce the ODE to a linear one, and solve:
1. $(x+1)(y’+y^2)=-y;$
2. $xy^2y’=^2+y^3;$
3. $xyy’=y^2+x.$
3. Reduce order of the equation, and solve: