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The general solution to a differential equation will contain unknown constants. We for example found that

has the solution

To start with we have a given constant, *a*, and then we have two constants we may need to find. That is the initial velocity and the initial position, and those two might be given. In that case, we say we have given initial conditions. We might also be given other conditions, like two points. Say we have that after 1 s we are at 10 m and at 5 s we are at 50 m, and that the acceleration is -10 m/s^{2}. We have

And the two points give us

or

That gives us

Solving this gives us

We thus get the particular solution

In general first order differential equations will have one constant of integration, a second order differential equation will have two, and so on.

**A second example**

Let us yet again look at

that we on the previous page found to have the solution

Let us find the solution that goes through the point (2, 3). We get

or 27=12+*C*, so *C*=15, and our particular solution will thus be

More in general, to find a solution that goes through the point (*x*_{0}, *y*_{0}) we substitute that into our general solution to get

or

This gives us the particular solution

or

You can try this in this Geogebra file. Grab and move point A.

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