Solve $\frac{dy}{dx}-\frac{1}{2}(1+\frac{1}{x})y+\frac{3}{x}{y}^{3}=0$?

Ciolan3u
2022-09-11
Answered

Solve $\frac{dy}{dx}-\frac{1}{2}(1+\frac{1}{x})y+\frac{3}{x}{y}^{3}=0$?

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I was trying to compute the solution for the following differential equation:

$x(2{x}^{2}ylog(y)+1){y}^{\prime}=2y$

As I couldn't get anywhere I checked the hints in the textbook which are the following:

Reverse the way of thinking, namely view $x$ as a function and $y$ as a variable, considering that

$y=\frac{dy}{dx}=\frac{1}{\frac{dx}{dy}}=>{y}^{\prime}=\frac{1}{{x}^{\prime}}$. Then it goes to say that the equation now becomes

${x}^{\prime}-\frac{x}{2y}=log(y){x}^{3}$

This final equation is obviously simple enough to solve, but how on Earth did they arrive there?

$x(2{x}^{2}ylog(y)+1){y}^{\prime}=2y$

As I couldn't get anywhere I checked the hints in the textbook which are the following:

Reverse the way of thinking, namely view $x$ as a function and $y$ as a variable, considering that

$y=\frac{dy}{dx}=\frac{1}{\frac{dx}{dy}}=>{y}^{\prime}=\frac{1}{{x}^{\prime}}$. Then it goes to say that the equation now becomes

${x}^{\prime}-\frac{x}{2y}=log(y){x}^{3}$

This final equation is obviously simple enough to solve, but how on Earth did they arrive there?

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