smetuwh
2022-09-30
Answered

How to you find the general solution of $\frac{dy}{dx}=x{e}^{{x}^{2}}$?

You can still ask an expert for help

Garrett Valenzuela

Answered 2022-10-01
Author has **9** answers

This is a First Order Separable DE, so we can just separate the variable in it's current form to give;

$\int dy=\int x{e}^{{x}^{2}}dx$

And note we can re-write as:

$\int dy=\frac{1}{2}\int 2x{e}^{{x}^{2}}dx$

So we can integrate to give:

$y=\frac{1}{2}{e}^{{x}^{2}}+c$

$\int dy=\int x{e}^{{x}^{2}}dx$

And note we can re-write as:

$\int dy=\frac{1}{2}\int 2x{e}^{{x}^{2}}dx$

So we can integrate to give:

$y=\frac{1}{2}{e}^{{x}^{2}}+c$

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This is a first order non-linear differential equation. Unfortunately, I have no experience solving non-linear differential equations. From the research I have done, this type of equation looks similar to the Ricatti equation. Is there a closed form solution to the above equation? How can I solve this equation? I'm interested in getting a function that shows how the pressure or velocity of the system decays with time.

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My question is to find the solutions to the following

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where ${f}^{-1}(x)$ refers to the inverse of the function f. The domain really isn't important, though I am interested in either (-inf, inf) or (0, inf), so if any solutions are known for more restricted domains then they are welcome.

I cannot find any material relating to this type of question in any of my calculus and differential equations textbooks and references; it seems quite unorthodox. Any material which covers this type of diff equation would be wlecome

$\frac{df(x)}{dx}={f}^{-1}(x)$

where ${f}^{-1}(x)$ refers to the inverse of the function f. The domain really isn't important, though I am interested in either (-inf, inf) or (0, inf), so if any solutions are known for more restricted domains then they are welcome.

I cannot find any material relating to this type of question in any of my calculus and differential equations textbooks and references; it seems quite unorthodox. Any material which covers this type of diff equation would be wlecome

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