Practicing Success
The equation of a curve passing through the origin and satisfying the differential equation $\frac{d y}{d x}=(x-y)^2$, is |
$e^{2 x}(1-x+y)=1+x-y$ $e^{2 x}(1+x-y)=1-x+y$ $e^{2 x}(1-x+y)+(1+x-y)=0$ $e^{2 x}(1+x+y)=1-x+y$ |
$e^{2 x}(1-x+y)=1+x-y$ |
We have, $\frac{d y}{d x}=(x-y)^2$ Let $x-y=v$. Then, $1-\frac{d y}{d x}=\frac{d v}{d x} \Rightarrow \frac{d y}{d x}=1-\frac{d v}{d x}$ ∴ $\frac{d y}{d x}=(x-y)^2$ $\Rightarrow 1-\frac{d v}{d x}=v^2$ $\Rightarrow 1-v^2 = \frac{d v}{d x}$ $\Rightarrow d x=\frac{1}{1-v^2} d v$ $\Rightarrow 2 \int d x=2 \int \frac{1}{1-v^2} d v$ $\Rightarrow 2 x=\log \left(\frac{1+v}{1-v}\right)+\log C$ $\Rightarrow C\left(\frac{1+v}{1-v}\right)=e^{2 x}$ $\Rightarrow C\left(\frac{x-y+1}{y-x+1}\right)=e^{2 x} \Rightarrow C(x-y+1)=e^{2 x}(y-x+1)$ Taking $C=1$, we find that option (a) is correct. |