Question

In Problems 1–22 solve the given differential equation by separation

of variables.

$\mathit{d}\mathit{y}\mathbf{-}{\mathbf{(}\mathbf{y}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{2}}\mathit{d}\mathit{x}\mathbf{=}\mathbf{0}$

Short answer

The solution of the given differential equation is$y=1-\frac{1}{x+C}$

Step-by-step solution

Step 1:Separable Equation

A first-order differential equation of the form $\frac{dy}{dx}=g\left(x\right)h\left(y\right)$is said to be separable or to have separable variables.

Separate the variables and integrate

The given equation is,$dy-{(y-1)}^{2}dx=0$.

Rearrange,

$dy={(y-1)}^{2}dx$

Separate the variables,

$\frac{dy}{{(y-1)}^2}=dx$

Rewrite,

${(y-1)}^{-2}dy=dx$

Integrate both sides,

$\begin{array}{l}\int {\left(Y-1\right)}^{2}dy=\int dx\\ \frac{-1}{y-1}=x+C\end{array}$

Solve fory,

$\begin{array}{l}-1=(y-1)(x+C)\\ -\frac{1}{x+C}=y-1\end{array}$

Simplify,

$y=1-\frac{1}{x+C}$

Hence, the solution of the given differential equation iswidth="104" style="max-width: none; vertical-align: -16px;" $\mathit{y}\mathbf{=}\mathbf{1}\mathbf{-}\frac{\mathbf{1}}{\mathbf{x}\mathbf{+}\mathbf{C}}$.