Differential Equations Question 296
Question: What is the solution of the differential equation $ a( x\frac{dy}{dx}+2y )=xy\frac{dy}{dx} $ -
Options:
A) $ x^{2}=ky{e^{\frac{y}{a}}} $
B) $ yx^{2}=ky{e^{\frac{y}{a}}} $
C) $ y^{2}x^{2}=ky{e^{\frac{y^{2}}{a}}} $
D) None of the above
Show Answer
Answer:
Correct Answer: D
Solution:
[d] Given differential equation is $ a( x\frac{dy}{dx}+2y )=xy\frac{dy}{dx} $
$ \Rightarrow ax\frac{dy}{dx}-xy\frac{dy}{dx}=-2ay $
$ \Rightarrow (xy-ax)\frac{dy}{dx}=2ay\Rightarrow x(y-a)\frac{dy}{dx}=2ay $
$ \Rightarrow x(y-a)dy=2aydx $
$ \Rightarrow \frac{(y-a)}{y}dy=\frac{2a}{x}dx\Rightarrow ( 1-\frac{a}{y} )dy=\frac{2a}{x}dx. $ $ dy-\frac{a}{y}dy=\frac{2a}{x}dx $ Integrate on both side $ \int{dy-a\int{\frac{1}{y}dy=2a\int{\frac{1}{x}dx}}} $ $ y-a\log y=2alogx+logc $
$ \Rightarrow y=a\log x^{2}yc\Rightarrow x^{2}y=k{e^{y/a}} $
BETA
