GATE Electrical 2026 | Question: 40

Question

Consider the second-order differential equation

\[
\frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}+y=0
\]
with initial conditions
\[
y(0)=1,\left.\quad \frac{d y}{d x}\right|_{x=0}=1
\]
The solution is given by

• A. $y(x)=\exp (-x / 2)\left(\cos \left(\frac{\sqrt{3} x}{2}\right)+\sqrt{3} \sin \left(\frac{\sqrt{3} x}{2}\right)\right)$
• B. $y(x)=\exp (-x / 2)\left(\cos \left(\frac{\sqrt{3} x}{2}\right)+\frac{1}{\sqrt{3}} \sin \left(\frac{\sqrt{3} x}{2}\right)\right)$
• C. $y(x)=\exp (-x / 2)\left(\cos \left(\frac{\sqrt{3} x}{2}\right)-\frac{1}{\sqrt{3}} \sin \left(\frac{\sqrt{3} x}{2}\right)\right)$
• D. $y(x)=\exp (-x / 2)\left(\cos \left(\frac{\sqrt{3} x}{2}\right)-\sqrt{3} \sin \left(\frac{\sqrt{3} x}{2}\right)\right)$