Solve 1st Order ODE from Transcendental Equation

[gonadas91]

It is a general doubt about the following equation: Imagine I want to calculate an unknown function $$y(x)$$, and my starting equation is of the type

$$y(x)^{2}=\frac{1}{x^{2}Log^{2}(A(x)y(x)^{2})}$$

, then, am I allowed to start with the equation

$$y(x)=\frac{1}{xLog(A(x)y(x)^{2})}$$

and differenciate in both sides of the equation, to obtain a first order diferential equation to get y(x)?

(Note that the beginning equation is a trascendental equation, but why not trying to solve the first order ODE?)Thank you

 

[Brage]

I don't see why this wouldn't be allowed but I'd assume setting $f(x)=y(x)^2$ and solving for $f(x)$ would be simpler. You can ofc use $log(a)^2=log(a)*log(a)=log(2a)$ to remove the square on the logarithmic function.

 

[gonadas91]

Mmm the last property of the logs is not valid. My question actually has to be with the modifications that are allowed on a differential equation. Also, the resulting differential equation using that is not depending on A(g) which annoys me somehow...

 

[Brage]

oh yeah ofc it isn't haha, too late for this here