Homework for Part 2
Week 3 : Solve exercises 1 to 3.
Week 4 : Solve exercises 4 to 6.
Exercise 1
Let \(f : \mathbb{R}\times \mathbb{R}\to \mathbb{R}\) a function and \(K \in \mathbb{R}\) a constant such that \[\left\vert\frac{\partial f}{\partial y}(t,y)\right\vert \leq K \qquad \forall (t,y) \in [a,b]\times[c,d].\]
Show that \[\vert f(t,y) -f(t,z) \vert \leq K \vert y-z \vert \qquad \forall (t,y), (t,z) \in [a,b]\times[c,d].\]
How to generalise this property for \(f: \mathbb{R}\times \mathbb{R}^n \to \mathbb{R}^n\) ?
Exercise 2
With no initial capital you are investing \(k\) CHF per year at an interest rate \(r\) compounded continuously.
Compute the sum \(S(t)\) accumulated at any time \(t.\)
If \(r=7.5\%,\) determine \(k\) in such a way you get 1 million of CHF in 40 years.
If \(k = 2000\) CHF, determine the interest rate that you have to get in order to have 1 million of CHF in 40 years.
Exercise 3
You invest for 10 years 1200 CHF at a rate of 2% per year.
Compute your capital after 10 years
when the interests are computed yearly.
when the interests are compounded continuously.
Exercise 4
Consider the equation \(\displaystyle\begin{cases} \dfrac{dN}{dt} = k (1-N)^2, \qquad k>0 \\ N(0) = N_0 >0 \end{cases} .\)
Show that \(N\) increases with \(t.\)
Sketch \(\dfrac{dN}{dt}\) versus \(N.\)
Solve the above initial value problem.
Show that 1 is a semistable equilibrium, i.e. \(N_0 < 1 \Rightarrow N(t) \to 1\) when \(t \to +\infty,\) and \(N_0 >1 \Rightarrow N(t)\) blows up.
Exercise 5
Consider the equation \(\displaystyle\begin{cases} \dfrac{dN}{dt} = N (1-N^2) \\ N(0) = N_0\in \mathbb{R}\end{cases} .\)
Find the stationary points of this equation.
Study their stability.
Exercise 6
Let \(f\) be a positive function such that \(\displaystyle\int_0^{+\infty} \dfrac{ds}{f(s)} < + \infty.\)
Show that the ordinary differential equation \(y' = f(y)\) for \(t>0\) and with \(y(0) = 0\) possesses a unique solution blowing up at \(\displaystyle t^{\ast} = \int_0^{+\infty} \dfrac{ds}{f(s)}.\)