$xy\prime -3y=x-1$ Solve

Mutignaniz2
2022-09-12
Answered

$xy\prime -3y=x-1$ Solve

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asked 2022-05-20

Recently I have met such an equation:

$x\frac{dy}{dx}=y+\sqrt{{x}^{2}+{y}^{2}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\frac{dy}{dx}-\frac{y}{x}=\frac{\sqrt{{x}^{2}+{y}^{2}}}{x}$

First of all what type it has? I can not refer it to any I am aware of(I am a newbie so it is likely that I'm missing something). Trying to solve it gave no result therefore I googled and youtube revealed that I had to use change of variables $y=v(x)x$. However it looks like a cheating. Why use this particular replacement and is there any way to solve it without it?

$x\frac{dy}{dx}=y+\sqrt{{x}^{2}+{y}^{2}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\frac{dy}{dx}-\frac{y}{x}=\frac{\sqrt{{x}^{2}+{y}^{2}}}{x}$

First of all what type it has? I can not refer it to any I am aware of(I am a newbie so it is likely that I'm missing something). Trying to solve it gave no result therefore I googled and youtube revealed that I had to use change of variables $y=v(x)x$. However it looks like a cheating. Why use this particular replacement and is there any way to solve it without it?

asked 2022-05-17

Solve the differential equatio

$\frac{dy}{dx}={\left(\frac{x+2y-3}{2x+y+3}\right)}^{2}$

The answer is given as:

$(x+3{)}^{3}-(y+3{)}^{3}=c(x-y+6{)}^{4}$

My attempt :I tried to expand the terms in the numerator and the denominator, arrange them into differentials and then integrate then, but I couldn't arrange all the terms in integrable format.

$\frac{dy}{dx}={\left(\frac{x+2y-3}{2x+y+3}\right)}^{2}$

The answer is given as:

$(x+3{)}^{3}-(y+3{)}^{3}=c(x-y+6{)}^{4}$

My attempt :I tried to expand the terms in the numerator and the denominator, arrange them into differentials and then integrate then, but I couldn't arrange all the terms in integrable format.

asked 2022-02-17

I am trying to solve a differential equation of the type $x{}^{\u2033}=-x+{x}^{3}$ . Now when I first integrate it with respect to time, $t$ , then I am getting ${x}^{\prime}=x({x}^{2}-1)+C$ , which is a non-linear first order differential equation. Now if the constant happens to be zero then I can solve it by partial fraction method but as life is not easier that constant term tagged to the $x}^{\prime$ is actually not zero. So how to proceed forward in this case.

Thank you!

Thank you!

asked 2022-09-11

What is a solution to the differential equation $xy\prime =y$?

asked 2022-07-10

I have a first order linear differential equation (a variation on a draining mixing tank problem) with many constants, and want to separate variables to solve it.

$\frac{dy}{dt}={k}_{1}+{k}_{2}\frac{y}{{k}_{3}+{k}_{4}t}$

y is the amount of mass in the tank at time t, and for simplicity, I've reduced various terms to constants, ${k}_{1}$ through ${k}_{4}$.

Separation of variables is made difficult by ${k}_{1}$, and I've considered an integrating factor, but think I might be missing something simple.

$\frac{dy}{dt}={k}_{1}+{k}_{2}\frac{y}{{k}_{3}+{k}_{4}t}$

y is the amount of mass in the tank at time t, and for simplicity, I've reduced various terms to constants, ${k}_{1}$ through ${k}_{4}$.

Separation of variables is made difficult by ${k}_{1}$, and I've considered an integrating factor, but think I might be missing something simple.

asked 2022-09-07

What is a solution to the differential equation $\frac{dy}{dx}=3$?

asked 2022-09-12

How can I solve the differential equation $y\prime =\mathrm{sin}x-x\mathrm{sin}x$?