Homework for Part 1
Week 1 : Solve exercises 1 to 4.
Week 2 : Solve exercises 5 to 8.
Exercise 1
Solve the following ordinary differential equations :
\(\displaystyle y' -2y = t^2 e^{2t}\)
\(\displaystyle t y' +2y = \sin(t)\) for \(t >0.\)
Exercise 2
Find the solution of the given initial value problem .
\(\displaystyle y' - 2y = e^{2t} \;\) with \(y(0) =2.\)
\(\displaystyle ty' + (t+1) y = t\;\) with \(y(\ln(2)) = 1.\)
Exercise 3
Show that the solution to \[\begin{cases} y' = y^{\alpha}, \quad \alpha >1, \\ y(0) = y_0 >0 \end{cases}\,\] blows up.
Exercise 4
Show that the initial value problem \[\begin{cases} y' = \sqrt{y}, \\ y(0) =0 \end{cases}\] admits infinitely many solutions.
Exercise 5
Solve \(\displaystyle y' = \frac{3t^2+4t+2}{2(y-1)}\) with initial condition \(y(0) = -1.\)
Exercise 6
Solve the equation \[\displaystyle t^2 y^3 + t (1+y^2)y' = 0\] using the integrating factor \(\displaystyle u = \frac{1}{ty^3}.\)
Exercise 7
Solve \(\displaystyle y' = \frac{t+3y}{t-y}.\)
Exercise 8 • Grönwall’s Inequality
Let \(y\) be a continuous function and \(L\in \mathbb{R}\) with \[0 \leq y(t) \leq L \int_0^t y(s) ds \qquad \text{for all} \; t>0.\]
Show that \(y \equiv 0\) for \(t\geq 0.\)