# Question about the uniqueness of solutions?

I was working on a few practice problems for my applied ordinary differential equations class and I was stumped on a really simple intro problem.

If we are told that a DFQ has the following form:

$$\frac{du}{dx} = f(u,x)$$

Is there any way to infer the number of solutions it has? Can $$u(x) = x^2$$ and $$u(x) = x^3$$ both be solutions? I'm thinking not, since usually a DFQ with multiple solutions has all solutions differ by a constant, not something with the dependent variable.

Are there standard rules for this?

Any help is extremely appreciated!

You can get both $$u=x^2$$ and $$u=x^3$$ as solutions if the exponent is the integration constant, that is, the general solution is $$u=x^C$$. Isolating $$C$$ and taking the derivative results in $$C=\frac{\ln u}{\ln x}\implies 0=\frac{u'}{u\ln x}-\frac{\ln u}{x(\ln x)^2}$$ so that the corresponding differential equation is $$u'=\frac{u\ln u}{x\ln x}.$$
You can also take other forms of interpolation like $$u=Cx^2+(1-C)x^3$$ so that $$C=\frac{u-x^3}{x^2-x^3}\implies 0=\frac{u'-3x^2}{x^2-x^3}-\frac{(u-x^3)(2x-3x^2)}{(x^2-x^3)^2}$$ leading to the differential equation $$u'=3x^2+\frac{(2-3x)(u-x^3)}{x(1-x)}$$