A) \[{{x}^{2}}=ky{{e}^{\frac{y}{a}}}\]
B) \[y{{x}^{2}}=ky{{e}^{\frac{y}{a}}}\]
C) \[{{y}^{2}}{{x}^{2}}=ky{{e}^{\frac{{{y}^{2}}}{a}}}\]
D) None of the above
Correct Answer: D
Solution :
| [d] Given differential equation is |
| \[a\left( x\frac{dy}{dx}+2y \right)=xy\frac{dy}{dx}\] |
| \[\Rightarrow ax\frac{dy}{dx}-xy\frac{dy}{dx}=-2ay\] |
| \[\Rightarrow (xy-ax)\frac{dy}{dx}=2ay\Rightarrow x(y-a)\frac{dy}{dx}=2ay\] |
| \[\Rightarrow x(y-a)dy=2aydx\] |
| \[\Rightarrow \frac{(y-a)}{y}dy=\frac{2a}{x}dx\Rightarrow \left( 1-\frac{a}{y} \right)dy=\frac{2a}{x}dx.\] |
| \[dy-\frac{a}{y}dy=\frac{2a}{x}dx\] |
| Integrate on both side |
| \[\int{dy-a\int{\frac{1}{y}dy=2a\int{\frac{1}{x}dx}}}\] |
| \[y-a\log \,y=2alogx+logc\] |
| \[\Rightarrow y=a\log {{x}^{2}}yc\Rightarrow {{x}^{2}}y=k{{e}^{y/a}}\] |
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