Question

If a curve passing through $\left(1 , 2\right)$ satisfies the differential equation $y\left(1 + x y\right)dx-xdy=0$ , then which of the following is true?

• A. $f\left(x\right)=\frac{2 x}{2 - x^{2}}$
• B. $f\left(x\right)=\frac{x + 1}{x^{2} + 1}$
• C. $f\left(x\right)=\frac{x - 1}{4 - x^{2}}$
• D. $f\left(x\right)=\frac{4 x}{1 - 2 x^{2}}$

Answer 1

Answer: (A)

The given differential equation is
$y\left(1 + x y\right)dx-xdy=0$
$\Rightarrow \left(y d x - x d y\right)+xy^{2}dx=0$
$\Rightarrow \frac{y d x - x d y}{y^{2}}+xdx=0$
$\Rightarrow d\left(\frac{x}{y}\right)+xdx=0$
On integrating both sides, we get
$\int d\left(\frac{x}{y}\right)+ \int xdx=0$
$\Rightarrow \frac{x}{y}+\frac{x^{2}}{2}=C$
If the above curve passes through the point $\left(1 , 2\right)$ , then
$\frac{1}{2}+\frac{1}{2}=C\Rightarrow C=1$
The given curve is $\frac{x}{y}+\frac{x^{2}}{2}=1$
$\therefore y=\frac{2 x}{2 - x^{2}}$
$\Rightarrow f\left(x\right)=\frac{2 x}{2 - x^{2}}$