Differential Equations Question 248
Question: The solution of the differential equation $ \sqrt{a+x}\frac{dy}{dx}+xy=0 $ is
[MP PET 1998]
Options:
A) $ y=A{e^{2/3(2a-x)\sqrt{x+a}}} $
B) $ y=A{e^{-2/3(a-x)\sqrt{x+a}}} $
C) $ y=A{e^{2/3(2a+x)\sqrt{x+a}}} $
D) $ y=A{e^{-2/3(2a-x)\sqrt{x+a}}} $ (Where A is an arbitrary constant.)
Show Answer
Answer:
Correct Answer: A
Solution:
Given $ \frac{dy}{dx}+\frac{xy}{\sqrt{a+x}}=0 $
Therefore $ \frac{dy}{y}=\frac{-xdx}{\sqrt{a+x}} $
Integrating both sides, $ \int{\frac{dy}{y}}=\int{\frac{-x}{\sqrt{x+a}}dx} $
$ \log y=-\int _{{}}^{{}}{\frac{x+a-a}{\sqrt{x+a}}}dx $
$ =-\int _{{}}^{{}}{\sqrt{x+a}}dx+\int _{{}}^{{}}{\frac{a}{\sqrt{x+a}}}dx $
Therefore $ \log y=-\frac{2}{3}{{(x+a)}^{3/2}}+2a\sqrt{x+a}+\log A $
$ y=A{e^{-2/3{{(x+a)}^{3/2}}+2a\sqrt{x+a}}} $
$ =A{e^{[ (\sqrt{x+a}( -\frac{2}{3}(x+a)+2a ) ]}} $
$ =A{e^{[ \sqrt{x+a}( \frac{-2x-2a+6a}{3} ) ]}} $
$ =A{e^{[-2/3\sqrt{x+a}(x-2a)]}} $
or $ y=A{e^{[2/3\sqrt{x+a}(2a-x)]}} $ .
BETA
