MATH 395 PROBLEMS 9

IGOR KRIZ

Regular problems:

1. Solve:

y′ = Ay

where

A = ( \begin{matrix}1 & 2 & 1 \\ 0 & 1 & −2 \\ 0 & 0 & 1 \end{matrix} )

2. Solve:

y′ = Ay + b(x)

where

A = ( \begin{matrix}2 & 3 \\ 0 & 2 \end{matrix} ) , b(x) = ( \begin{matrix}x \\ 1 \end{matrix} ) .

3. Using Fubini’s theorem, calculate ∫_{[0,1]×[0,1]} xy(x + y)dxdy.

4. Using Fubini’s theorem, evaluate

∫_{[0,t]×[1,t]} y^{−3}e^{tx/y}dxdy,

where t > 1.

Challenge problem:

5. Let I = [a_{1}, b_{1}]×⋯×[a_{n}, b_{n}]. Prove that if f is bounded on I and continuous on (a_{1}, b_{1}) × ⋯ × (a_{n}, b_{n}), then the Riemann integral

∫_{I} f

exists. [Generlize the case n = 1 which was done in class at the beginning of LDE’s.]