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(This page basically follows my train of thoughts, so it is not all too well structure, but that is on purpose – I want you to get a feeling of the thoughts involved.)

Solving differential equations can be quite hard, and even a small change to an equation can make it all but impossible to solve. We previously found that

has the solution

Let us now instead look at

This is a linear first order equation that is not separable, so we may try with using an integration factor. We get

This gives us

or

Then we integrate to get

We may try to solve the RHD integral by integration by parts. If we try directly we will find that we cannot integrate the exponential function, and integrating the *x*^{2} part will just increase the power. Here we will use a little trick and split the function in another way to get

And here we are basically stuck because the integral cannot be solve with elementary functions. So our answer, for the moment being, is

or

All is not lost though, because say we want to have a solution that goes through (1, 2) and we want to know its value when *x*=2. We could write our solution as

Where *a* is a constant. Next we substitute in our coordinates

so

So we need to find a way to find this value *a*. How to do this? Yet again all is not lost. If you remember that the cumulative standard normal probability function is

and most scientific calculators and mathematical software packages can calculate this (or something called the error function). So

This, by the way, allow us to write the solution to our differential equation as

Ok, so we do have a general solution, but yet again, say we need a particular solution that passes through a point (*x*_{0}, *y*_{0})? We get

or in our case (1, 2), so

So we have the particular solution

Back to the question, what is *y* when *x*=2? Substituting 2 into our equation we get about 13.96 for *y*.

A general solution that will go through a point (*x*_{0}, *y*_{0}) will thus be

This can be tested with this Texas instruments Nspire file: ypxyisx2

or this Geogebra file.

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