Question

Solution of the differential equation: $\frac{x+y\frac{dy}{dx}}{y-x\frac{dy}{dx}}=\frac{xsi{n}^2(x^2+y^2)}{y^3}$

• A. $-cot(x^2+y^2)={\left(\frac{x}{y}\right)}^2+c$
• B. $\frac{y^2}{x^2+y^{2}c}=-ta{n}^2(x^2+y^2)$
• C. $-cot(x^2+y^2)={\left(\frac{y}{x}\right)}^2+c$
• D. $\frac{y^2+x^{2}c}{x^2}=-ta{n}^2(x^2+y^2)$

Answer 1

Answer: (A), (B)

The correct options are
A $-cot(x^2+y^2)={\left(\frac{x}{y}\right)}^2+c$
B $\frac{y^2}{x^2+y^{2}c}=-ta{n}^2(x^2+y^2)$

The given differential equation can be written as
$\frac{xdx+ydy}{ydx-xdy}=\frac{xsi{n}^2(x^2+y^2)}{y^3}\Rightarrow \frac{2xdx+2ydy}{ydx-xdy}=\frac{2xsi{n}^2(x^2+y^2)}{y^3}\Rightarrow \frac{d(x^2+y^2)}{si{n}^2\left(x^2{y}^2\right)}=\frac{2x}{y^3}(ydx-xdy)\Rightarrow cose{c}^2(x^2+y^2)d(x^2+y^2)=2\left(\frac{x}{y}\right)d\left(\frac{x}{y}\right)$
On integrating, we get
$\int cosc{e}^2(x^2+y^2)d(x^2+y^2)=2\int \left(\frac{x}{y}\right)d\left(\frac{x}{y}\right)$
$-cot(x^2+y^2)={\left(\frac{x}{y}\right)}^2+c$
$\frac{y^2}{x^2+y^{2}c}=-ta{n}^2(x^2+y^2)$