Question

The differential equation of the family of lines which pass through $(1,-1)$ is:

• A. $y=(x+1)\frac{dy}{dx}+1$
• B. $y=(x+1)\frac{dy}{dx}-1$
• C. $y=(x-1)\frac{dy}{dx}+1$
• D. $y=(x-1)\frac{dy}{dx}-1$

Answer 1

The correct option is D $y=(x-1)\frac{dy}{dx}-1$

The gradient of a line $=\frac{dy}{dx}$

$\therefore$ The equation of the line passing through $(1,-1)$ is

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

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

$\Rightarrow \log (x-1)=\log (y+1)-\log c$

$\Rightarrow \log (y+1)=\log \left(c(x-1)\right)$

$\Rightarrow y=x(x-1)-1$