I'm reading Morris Tenenbaum's book and it says that given a solvable n-th order differential equation, we can have more than one n-parameter family of solutions.
By n-parameter family of solution he means what is usually called a "general solution" , a possible solution with n coeficients.He calls it n-parameter family of solutions because it is not necessarily a general solution ( it might not contain some possible particular solutions ).
He gives many examples :
Given the first order ODE $y' = -2y^{3/2}$ while we might obtain the following 1-parameter family of solutions $y = \frac{1}{(x+c)^2}$ , it's not the only possible one.
There's also the 1-parameter family of solutions $y = \frac{C^2}{(Cx + 1)^2}$ .
While the particular (or singular ) solution $y=0$ for example, can't be obtained by the first 1-parameter family of solutions ( no matter what $C$ we choose ), it can be obtained by the second 1-parameter family of solutions ( by setting $C = 0$ ).
The problem i'm having :
My doubt is if, even tho a single n-parameter family of solutions might not provide all possible particular solutions, any particular solution can be obtained by any n-parameter family of solutions.
For example, given the first order ODE $y = xy' + (y')^2$
We can easily find one possible 1-parameter family of solutions as $y = cx + c^2$.
But it also has a singular solution $y = -\frac{x^2}{4}$ that can't be obtained by $y = cx + c^2$ ( no matter what c we choose ).
My question is : Could $y = -\frac{x^2}{4}$ be obtained by another 1-parameter family of solutions
that would satisfy $y = xy' + (y')^2$ ? Is it a rule that there's always is a n-parameter family of solutions that can provide any particular solution ?
Best Answer
I don't think there is a global rule in which you can reach the target as you wanted unless we have some certain restrictions. The OE $y = xy' + (y')^2$ is a Clairaut's equation and you know that in this case we encounter a new creature called envelope. If you have a look at the documents about an envelope, you'll find sometimes we have any singly infinite system of curves exactly as you indicated as $y=cx+c^2$above. We also note that these 1-parameter family of solutions have always a fixed curve which touches any curves of family in just one point. So it can't be for us to find this fixed one from the family so it is singular. Note that there is a thin difference between an $n-$ family of solutions and a general solution. We call the $n-$ parameter family of solutions, a general solution iff it can contain every particular solutions. These two definition are coincided for example when we work with linear OE with constant coefficients. There we have strong theorem which guarantee this point. For more information see Differential Equation by H.T.H.Piaggio or in your book p.35 or theorem 19.3 p 208.