Question

For the differential equation xy $$\frac{dy}{dx}$$ = (x + 2) (y + 2). Find the solution curve passing through the point (1, −1).

Answer 1

We have,

$$xy\frac{dy}{dx}=(x+2)(y+2)$$

$$\Rightarrow \frac{y}{y+2}dy=\frac{(x+2)}{x}dx$$

Integrating both sides, we get

$$\int \frac{y}{y+2}dy=\int \frac{(x+2)}{x}dx$$

$$\Rightarrow \int dy-2\int \frac{1}{y+2}dy=\int dx+2\int \frac{1}{x}dx$$

$$\Rightarrow y-2\log |y+2|=x+2\log \left|x\right|+C.....(1)$$

This equation represents the family of solution curves of the given differential equation.

We have to find a particular member of the family, which passes through the point (1, - 1).

Substituting x = 1 and y = - 1 in (1), we get

$$-1-2\log \left|1\right|=1+2\log \left|1\right|+C$$

$$\Rightarrow C=-2$$

Putting ${\displaystyle C=-2}$ in (1), we get

$$y-2\log |y+2|=x+2\log \left|x\right|-2$$

$$\Rightarrow y-x+2=\log \left\{x^2{(y+2)}^2\right\}$$

$$\text{Hence, }y-x+2=\log \left\{x^2{(y+2)}^2\right\}\text{ is the equation of the required curve.}$$