Question

The solution of the differential equation $\sqrt{a+x}\frac{dy}{dx}+xy=0$ (Where A is an arbitrary constant.)

• 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. $y=A{e}^{-2/3(2a-x)\sqrt{x+a}}$

Answer 1

The correct option is A $y=A{e}^{2/3(2a-x)\sqrt{x+a}}$


Given $\frac{dy}{dx}+\frac{xy}{\sqrt{a+x}}=0\Rightarrow \frac{dy}{y}=\frac{-xdx}{\sqrt{a+x}}$
Integrating both sides, $\int \frac{dy}{y}=\int \frac{-x}{\sqrt{x+a}}dx$
$logy=-\int \frac{x+a-a}{\sqrt{x+a}}=-\int \sqrt{x+a}dx+\int \frac{a}{\sqrt{x+a}}dx\Rightarrow logy=-\frac{2}{3}(x+a{)}^{3/2}+2a\sqrt{x+a}+logAy=A{e}^{-2/3(x+a{)}^{3/2}+2a\sqrt{x+a}}=A{e}^{\left[\sqrt{x+a}(-\frac{2}{3}(x+a)+2a)\right]}=A{e}^{\left[\sqrt{x+a}\left(\frac{-2x-2a+6a}{3}\right)\right]}=A{e}^{[-2/3\sqrt{x+a}(x-2a\left)\right]}$
or $y=A{e}^{\left[2/e\sqrt{x+a}\right(2a-x\left)\right]}$