A) \[y=\{\phi (x)-1\}+c{{e}^{-\phi (x)}}\]
B) \[y\phi (x)={{\{\phi (x)\}}^{2}}+c\]
C) \[y{{e}^{\phi (x)}}=\phi (x){{e}^{\phi (x)}}+c\]
D) None of these
Correct Answer: A
Solution :
| [a] We have, \[dy+\{y\phi '(x)-\phi (x)\phi '(x)\}dx=0\] |
| \[\Rightarrow \frac{dy}{dx}+\phi '(x)\cdot y=\phi (x)\phi '(x)\] ? (i) |
| This is linear differential equation with |
| I.F \[={{e}^{\int{\phi '(x)dx}}}={{e}^{\phi (x)}}\] |
| Multiplying Eq. (i) by \[\phi (x)\] and integrating, we get |
| \[y{{e}^{\phi (x)}}=\int{\phi (x)\phi '(x){{e}^{\phi (x)}}}dx\] |
| \[\Rightarrow \,\,\,y{{e}^{\phi (x)}}=\int{{{e}^{\phi (x)}}\phi (x)\phi '(x)}dx\] |
| \[\Rightarrow y{{e}^{\phi (x)}}=\int{\phi (x){{e}^{\phi (x)}}}\phi '(x)dx\] |
| \[\Rightarrow y{{e}^{\phi (x)}}=\phi (x){{e}^{\phi (x)}}-\int{\phi '(x){{e}^{\phi (x)}}}dx\] |
| \[\Rightarrow y{{e}^{\phi (x)}}=\phi (x){{e}^{\phi (x)}}-{{e}^{\phi (x)}}+c\] |
| \[\Rightarrow y=\{\phi (x)-1\}+c{{e}^{-\phi (x)}}\] |
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