Question

Solution of $\frac{d^{2}y}{dx^{2}} + 4\frac{dy}{dx} + 29y = 0$, $y(0) = 0$ is}

• A. $y = ce^{-2x} \cos 5x$
• B. $y = ce^{-2x} \sin 2x$
• C. $y = ce^{-2x} \sin 5x$
• D. $y = ce^{2x} \cos 5x$

Hint:

If the roots are $\alpha \pm i\beta$, the exponential part uses $\alpha$ and the trigonometric part uses $\beta$.

Answer 1

The Correct Option is C

Step 1: Concept
This is a second-order linear homogeneous differential equation with constant coefficients. The auxiliary equation is $m^{2} + 4m + 29 = 0$.

Step 2: Meaning
Solving for $m$ using the quadratic formula: $m = \frac{-4 \pm \sqrt{16 - 116}}{2} = \frac{-4 \pm 10i}{2} = -2 \pm 5i$.

Step 3: Analysis
For complex roots $\alpha \pm i\beta$, the general solution is $y = e^{\alpha x}(c_{1} \cos \beta x + c_{2} \sin \beta x)$. Here, $\alpha = -2$ and $\beta = 5$, so $y = e^{-2x}(c_{1} \cos 5x + c_{2} \sin 5x)$.

Step 4: Conclusion
Applying the initial condition $y(0) = 0$: $0 = e^{0}(c_{1} \cos 0 + c_{2} \sin 0) \Rightarrow c_{1} = 0$. Thus, the solution simplifies to $y = c_{2}e^{-2x} \sin 5x$, which matches the form in option (C). Final Answer: (C)