MATH 396 PROBLEMS 1 REVIEW PROBLEMS

IGOR KRIZ

Regular problems:

1. Investigate the existence and convergence of the following (Lebesgue) integral:

∫_{−∞}^{∞} \frac{xdx}{(1 + x^{2})^{2}\sqrt{|x − 1|}}.

2. Solve: y′′ + y′ + y = x.

3. Solve:y′ = \frac{y+1}{x+2} + 1.

4. Can the following functions be the characteristics of a differential equation

y′ = f(x, y)

where f is continuous and \frac{∂f}{∂y} exists and is continuous everywhere? (a) y = x + c

(b) y = cx + 1.

Challenge problem:

5. Compute:

∫_{[0,1]^{n}} max(x_{1}, ..., x_{n})dx_{1}...dx_{n}.