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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!

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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 $$ u'=3x^2+\frac{(2-3x)(u-x^3)}{x(1-x)} $$

etc.

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  • $\begingroup$ That makes so much sense. Thank you! I don't believe we've learned about interpolation yet, but I'm sure we will. $\endgroup$ – Ced Feb 5 at 17:42

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