Exercise 3.1

1. We look for the solution as a power series of the form

   
   

   Imposing the condition that x satisfies the equation, we deduce

   
   

   From the initial condition and identifying the coefficients, we get

   
   

   from where it follows

   
   

   the series being uniformly and absolutely convergent on [ ?1, 1].

2. Proceeding as in the preceding exercise, we find

   
   

   the series being absolutely convergent on and uniformly convergent on every compact interval.

3. Similarly, we have

   
   

   the series being absolutely convergent on and uniformly convergent on every compact interval.

4. The solution of the equation is

   
   

   for every , i.e.

   
   

5. The solution of the equation is

   
   

   for every , i.e. x(t)=e ^t for every .

6. The solution of the equation is

   
   

   for every , i.e. for every .

Problem 3.1 We look for x as . Asking that x satisfy the equation, we obtain . By identifying the coefficients, we get

for every . From these equalities, we deduce

for every . Taking successively ( c ₀, c ₁)=(1, 0) and ( c ₀, c ₁)=(0, 1), we get the solutions

and

both series being absolutely convergent on , and uniformly convergent on every compact interval. Since the Wronskian of this system of solutions is nonzero at t=0, it follows that { x ₁, x ₂} is a fundamental system of solutions for the Hermite equation. If , one can easily see that one of the two...