Homework One

1

$$y(x)=\frac{x^2}{2}e^{-x}+1+C e^{-x}.$$

2

$$y(x)=-\frac{\cos x}{x} + \frac{\sin x}{x^2} + \frac{C}{x^2}.$$

3

$$y(x)=(x^2+C)e^{-x^2}.$$

4

$$y(x) = \frac{\arctan x +C}{(1+x^2)^2}.$$

5

$$y(x)=\frac{\sin x}{x^2}.$$

6

$$y(x) = \frac{C-\cos x}{x^3}.$$

7

$$y(x) = \frac{e^x +1 - e}{x}.$$

8

$$y(x) = e^{-x}\int\limits_0^x \frac{e^t}{1+t^2} d t.$$

9

$$y(x) = x^2+ C x. \lim\limits_{x\to 0}y(x) = 0, {\rm for all C.}$$