Question

The solution of the differential equation \[ \sqrt{a+x}\,\frac{dy}{dx} + xy = 0 \] is

• A. \(y = Ce^{\frac{2}{3}(2a-x)\sqrt{x+a}}\)
• B. \(y = Ce^{\frac{2}{3}(a-x)\sqrt{x+a}}\)
• C. \(y = Ce^{\frac{2}{3}(2a+x)\sqrt{x+a}}\)
• D. \(y = Ce^{-\frac{2}{3}(2a-x)\sqrt{x+a}}\)

Hint:

Use substitution \(t=a+x\) whenever \(\sqrt{a+x}\) appears—it simplifies integration instantly.

Answer 1

The Correct Option is A

Step 1: Rewrite equation.
\[ \sqrt{a+x}\frac{dy}{dx} + xy = 0 \Rightarrow \frac{dy}{dx} = -\frac{x}{\sqrt{a+x}}y \]

Step 2: Separate variables.
\[ \frac{dy}{y} = -\frac{x}{\sqrt{a+x}}dx \]

Step 3: Substitute.
Let \(t = a+x \Rightarrow x = t-a,\ dx = dt\) \[ \int \frac{dy}{y} = -\int \frac{t-a}{\sqrt{t}}dt \] \[ = -\int \left(t^{1/2} - a t^{-1/2}\right)dt \]

Step 4: Integrate.
\[ \log y = -\left(\frac{2}{3}t^{3/2} - 2a t^{1/2}\right) + C \] \[ = -\frac{2}{3}(a+x)^{3/2} + 2a\sqrt{a+x} + C \]

Step 5: Simplify.
\[ \log y = \frac{2}{3}(2a - x)\sqrt{a+x} + C \] \[ \Rightarrow y = Ce^{\frac{2}{3}(2a-x)\sqrt{a+x}} \] Conclusion: \[ {(A)} \]