ORDINARY DIFFERENTIAL EQUATIONS
1. Given an ordinary differential equations with separable variables y ′ = a(t)b(y), with a(t) defined on an
interval I and b(y) defined on an interval J, write the definition of solution of the differential equation.
2. Consider the differential equation y ′ = t3 b(y), with b(y) of class C 1 in R, such that b(y) > 0, for all t ∈ R.
(a) Prove that all the solutions defined in a neighborhood of t = 0 have a minimum at t = 0.
(b) If b(y) is also increasing in R, prove that all the solutions defined in a neighborhood of t = 0 are
convex for t > 0.
3. Write an example of an ordinary differential equations with separable variables
(a) that has no constant solutions;
(b) that has exactly two constant solutions;
(c) whose solutions are strictly increasing.
4. Consider the second order differential equation
x′′ + a(t)x′ + b(t)x = 0.
(a) Write the definition of solution of the differential equation.
(b) Write a Caucly problem for the given equation, then provide the definition of solution of the given
Cauchy problem.
5. Consider the second order differential equations
(1) x′′ + a(t)x′ + b(t)x = f (t)
(2) x′′ + a(t)x′ + b(t)x = g(t),
where the functions a, b, f, g are continuous in R. Let φ(t), ψ(t) be respectively solutions of the equations
(1) and (2).
Prove the so called Superposition Principle, that is that φ(t)+ψ(t) is a solution of the differential equation
x′′ + a(t)x′ + b(t)x = f (t) + g(t).
⃝2017
c Politecnico di Torino
