Proceed as in this example and find an explicit solution of the given initial-value problem. dy dx = y2 sin(x2), y(−4) = 1 6

is separable as

Integrating both sides gives

The integral on the right unfortunately has no elementary result (i.e. not in terms of Fresnel integrals). But we are given an initial value, so we can use the fundamental theorem of calculus to find a solution in terms of an integral.

Recall that

where .

Let be the antiderivative of . The theorem allows us to write

So the solution to the ODE is (upon integrating both sides)

Just to confirm this is valid, differentiating both sides gives

so all is well.

We can go on to solve for directly by taking the reciprocal of both sides, ending up with

what is the answer?

answer: the sun

explanation:

water cycle describes the evaporation of water from earth's surface, formation of clouds and precipitation water back onto the surface of earth. seventy percent of the earth is covered with water in the form of seas and oceans. the heat of the sun causes the evaporation of water from the surface of the ocean. the water vapors rise and condense to form water droplets. the collection of these water droplets form clouds which rain down again.

the evaporation of the oceans surface water as part of the water cycle is caused by the sun