How to you find the general solution of $\sqrt{1-4{x}^{2}}y\prime =x$?

engausidarb
2022-09-16
Answered

How to you find the general solution of $\sqrt{1-4{x}^{2}}y\prime =x$?

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Find the general solution of the first-order linear differential equation
$(dy/dx)+(1/x)y=6x+2$ , for x > 0

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

asked 2022-05-27

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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Find the derivative of the function.

$f\left(\theta \right)=\mathrm{cos}\left({\theta}^{2}\right)$

asked 2020-11-10

Solve differential equation
$(2y-{e}^{x})dx+xdy=0$ , x>0

asked 2022-04-12

What method would you use to solve:

$(1+{x}^{2})\frac{\mathrm{d}y}{\mathrm{d}x}=1+{y}^{2}\phantom{\rule{thickmathspace}{0ex}};\phantom{\rule{2em}{0ex}}y(2)=3$

I am asking this because I only know two methods of solving the DEs - separation of variables and integrating factor. Since the separation of variables does not work here, I tried integrating factor, however, I don't know what to do with the y2, because for the IF to work I need to get y on its own $\frac{\mathrm{d}y}{\mathrm{d}x}+P(x)y=Q(x)$)

What method do I use to solve this?

$(1+{x}^{2})\frac{\mathrm{d}y}{\mathrm{d}x}=1+{y}^{2}\phantom{\rule{thickmathspace}{0ex}};\phantom{\rule{2em}{0ex}}y(2)=3$

I am asking this because I only know two methods of solving the DEs - separation of variables and integrating factor. Since the separation of variables does not work here, I tried integrating factor, however, I don't know what to do with the y2, because for the IF to work I need to get y on its own $\frac{\mathrm{d}y}{\mathrm{d}x}+P(x)y=Q(x)$)

What method do I use to solve this?

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Solve differential equation