# What is a general solution to a differential equation?

I understand how to find the general solution of a differential equation, for complex roots, simple roots, nonhomogeneous equations etc... but I still don't really know what the general solution is?

As in, what is the purpose of it? What is it used for? thanks...

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• To say that you have found the general homogeneous solution means that this function solves the homogeneous equation for every choice of the constant $C_1$ and every solution of the homogeneous equation is of this form for some choice of $C_1$.

• You can actually have more than one particular solution to a DEQ. The difference between any two particular solutions is always a homogeneous solution.

Example:

$$y' + \left(\frac{a}{t}\right)y = t^3$$

The homogeneous solution is:

$$\displaystyle y_H = c_1 t^{-a}$$

Here are two particular solutions:

$$\displaystyle y_{1P} = \frac{t^4}{4+ a}$$

$$y_{2P} = \displaystyle \frac{t^4}{4+ a} + c_1 t^{-a}$$

What is the difference between these two particular solutions?

• To say you have a unique solution means that this is the ONLY function that satisfies both the differential equation and the initial condition. The graph of this function is the only solution curve that passes through the initial point.

For this, we obviously need to be given an initial condition.

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really well-answered! –  amWhy Apr 18 '13 at 1:11
indeed it was! thanks Amzoti –  Johnathon Svenkat Apr 18 '13 at 1:20
@JohnathonSvenkat: You are very welcome and it is GREAT to always question these things, as over time, we all forget these things because we learn to take them for granted from just hammering out solutions! Regards –  Amzoti Apr 18 '13 at 1:23