Integrals as solutions

Solve $\frac{dy}{dx} = e^x + x$ and $y(0) = 10$.

Solve $x' = \frac{1}{x^2}$, $x(1)=1$.

Solve $x' = \frac{1}{\cos(x)}$, $x(0)=\frac{\pi}{4}$.

Sid is in a car traveling at speed $10t+70$ miles per hour away from Las Vegas, where $t$ is in hours. At $t=0$, Sid is 10 miles away from Vegas. How far from Vegas is Sid 2 hours later?

Solve $y' = y^n$, $y(0) = 1$, where $n$ is a positive integer. Hint: You have to consider different cases.

The rate of change of the volume of a snowball that is melting is proportional to the surface area of the snowball. Suppose the snowball is perfectly spherical. The volume (in centimeters cubed) of a ball of radius $r$ centimeters is $(\nicefrac{4}{3}) \pi r^3$. The surface area is $4 \pi r^2$. Set up the differential equation for how the radius $r$ is changing. Then, suppose that at time $t=0$ minutes, the radius is 10 centimeters. Aft…

Find the general solution to $y''''= 0$. How many distinct constants do you need?