Solve the given differential equation by separation of variables. dy/dx = (x - 1)/(y + 2).

Solution:

Given, the differential equation is dy/dx = (x - 1)/(y + 2)

We have to solve the differential equation by separation of variables.

Now, multiply both sides by dx

dy = (x - 1)/(y + 2) dx

(y + 2)dy = (x - 1)dx

By integrating,

\(\int (y+2)dy=\int (x-1)dx\)

\((\frac{y^{2}}2{+2y})=(\frac{x^{2}}{2}-x)+C\)

\(\frac{y^{2}}{2}+2y-\frac{x^{2}}{2}+x-C=0\)

Multiplying by 2,

\(y^{2}+4y-x^{2}+4x-2C=0\)

Therefore, the general solution of the differential equation is \(y^{2}+4y-x^{2}+4x-2C=0\)

Solve the given differential equation by separation of variables. dy/dx = (x - 1)/(y + 2).

Summary:

The solution of the given differential equation dy/dx = (x - 1)/(y + 2) by separation of variables is \(y^{2}+4y-x^{2}+4x-2C=0\)