What is a general solution to a differential equation?

• 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.