Question
Solution of the differential equation $$x = 1 + xy\frac{{dy}}{{dx}} + \frac{{{x^2}{y^2}}}{{2!}}{\left( {\frac{{dy}}{{dx}}} \right)^2} + \frac{{{x^3}{y^3}}}{{3!}}{\left( {\frac{{dy}}{{dx}}} \right)^3} + ......$$
A.
$$y = \ln \left( x \right) + c$$
B.
$$y = {\left( {\ln \,x} \right)^2} + c$$
C.
$$y = \pm \ln \left( x \right) + c$$
D.
$$xy = {x^y} + c$$
Answer :
$$y = \pm \ln \left( x \right) + c$$
Solution :
The given equation is reduced to
$$\eqalign{
& x = {e^{xy\left( {\frac{{dy}}{{dx}}} \right)}} \cr
& \Rightarrow \ell n\,x = xy\frac{{dy}}{{dx}} \cr
& \Rightarrow \int {y\,dy} = \int {\frac{1}{x}} \ell n\,x\,dx \cr
& \Rightarrow \frac{{{y^2}}}{2} = \frac{{{{\left( {\ell n\,x} \right)}^2}}}{2} + c \cr
& \Rightarrow y = \pm \sqrt {{{\left( {\ell n\,x} \right)}^2}} + c \cr
& \Rightarrow y = \pm \ell n\,x + c \cr} $$