I'm having trouble learning differential equations. I'm a bit confused about the difference between a general solution of a differential equation and a family of solutions to a differential equation. I've repeatedly read obscure textbooks and concluded the following:
The general solution of a differential equation refers to the set of all solutions that satisfy the differential equation, and it usually contains some arbitrary constants, which can be determined according to the initial conditions or boundary conditions. For example, the general solution to the differential equation $y′=y$ is $y=Ce^{x}$, where $C$ is an arbitrary constant.
a family of solutions to a differential equation refers to a class of special solutions that satisfy differential equations, and there is a certain relationship between them, usually in the form of parameterization. For example, the family of differential equations $y′=y$ is $y=ae^{x+b}$, where a and b are parameters
(1)The general solution of differential equations contains all the solutions satisfying the differential equations, while the family of solutions to a differential equation to a differential equation contains only a part of the solutions satisfying the differential equations.
(2)Any constant in the general solution of differential equations can determine the unique solution through initial conditions or boundary conditions, but the parameters in the family of solutions to a differential equation cannot determine the unique solution through initial conditions or boundary conditions, but need additional parameter values.
(3)The family of solutions to a differential equation can be obtained by some transformation of the general solution of differential equations, but not necessarily vice versa
May I ask if there are any mistakes in the summary of the above three points?
Best Answer
In mathematics like in many other areas there is the phenomenon that we use the same word for different things and different words for the same thing. Some is the result from designing a consistent naming practice, some is that things that were historically different are not considered so any longer, and some is highlighting different aspects of some object.
In the case discussed here there is barely a difference. In my opinion,
Also, again in my opinion, "integration constant" is slightly more associated to "general solution", while "initial value" goes more along with "solution family".
But in the end, if you want to present a solution family, you do that with one or multiple solution formulas with parameters.
Note that for instance in Riccati equations, the solutions are parametrized by a projective circle. If that is reduced to a real parameter, the point at infinity is lost, so from that general solution to the full solution family is a difference of one solution.
At the fringe you have the Clairaut equations, where you get a solution family of lines and a singular solution that is tangent to this family. Also you can switch from the lines to the singular solution and back. I'm not sure what one would consider "general solution" here.