Differential Equations Question 254
Question: Solution of the differential equation $ x=1+xy\frac{dy}{dx}+\frac{x^{2}y^{2}}{2!}{{( \frac{dy}{dx} )}^{2}}+ $
$ \frac{x^{3}y^{3}}{3!}{{( \frac{dy}{dx} )}^{3}}+………… $
Options:
A) $ y=ln(x)+c $
B) $ y={{(lnx)}^{2}}+c $
C) $ y=\pm ln(x)+c $
D) $ xy=x^{y}+c $
Show Answer
Answer:
Correct Answer: C
Solution:
[c] The given equation is reduced to $ x={e^{xy(dy/dx)}} $
$ \Rightarrow \ell nx=xy\frac{dy}{dx}\Rightarrow \int{ydy=\int{\frac{1}{x}\ell nxdx}} $
$ \Rightarrow \frac{y^{2}}{2}=\frac{{{(\ell nx)}^{2}}}{2}+c $
$ \Rightarrow y=\pm \sqrt{{{(\ell nx)}^{2}}}+c=\pm \ell nx+c $
BETA
