Adela Brown

Answered

2021-12-22

Find a solution of $x\frac{dy}{dx}={y}^{2}-y$ that passes through the indicated points.

a.$(0,0)$

b.$(\frac{1}{4},\frac{1}{4})$

c.$(6,\frac{1}{8})$

a.

b.

c.

Answer & Explanation

Melissa Moore

Expert

2021-12-23Added 32 answers

By separing variables in $x\frac{dy}{dx}={y}^{2}-y$ ; we have

$\frac{dy}{{y}^{2}-y}=\frac{dx}{x}$

$\frac{dy}{y(y-1)}=\frac{dx}{x}$

$\left[\frac{-1}{y}\right]+\frac{1}{y-1}]dy=\frac{dx}{x}$

On integration, we obtain

$-\mathrm{ln}\left(y\right)+\mathrm{ln}(y-1)=\mathrm{ln}\left(x\right)+\mathrm{ln}c$

$\mathrm{ln}\left(\frac{y-1}{y}\right)=\mathrm{ln}\left(xc\right)$

$\frac{y-1}{y}=xc$

$y-1=xyc$

$y-xyc$

$y(1-xc)=1$

$y=\frac{1}{1-xc}$

$y=\frac{1}{1-cx}$ is a solution and $y=0$ is another constant solution

a) At$(0,0)$

$y=\frac{1}{1-xc}$

$0=\frac{1}{1-0}$

$0=1$

This is not possible

Therefore in this case there is no solution.

On integration, we obtain

a) At

This is not possible

Therefore in this case there is no solution.

karton

Expert

2021-12-30Added 439 answers

c) At

Here,

Substitute

Thus,

The solution of the initial value problem is

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