Consider the differential equation $\frac{d^2 y}{dx^2} + 8\frac{dy}{dx} + 16y = 0$ with boundary conditions $y(0) = 1$ and $\frac{dy}{dx}(0) = 0$. The solution to this equation is:
Options:
(A) $y = (1 + 2x)e^{4x}$
(B) $y = (1 - 4x)e^{4x}$
(C) $y = (1 + 8x)e^{-4x}$
(D) $y = (1 + 4x)e^{-4x}$
Official Answer Key:
D
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2021 Q.11)**
**1. Intuitive Physical Setup & Given Parameters**:
We are given the 2nd-order homogeneous linear ODE with constant coefficients:
\frac{d^2 y}{dx^2} + 8\frac{dy}{dx} + 16y = 0
subject to initial conditions $y(0) = 1$ and $y'(0) = 0$.
**2. Step-by-Step Mathematical Derivation**:
1. **Characteristic Equation**:
Assuming a solution of the form $y = e^{rx}$, the characteristic equation is:
r^2 + 8r + 16 = 0 \implies (r + 4)^2 = 0
This yields real, repeated roots: $r_1 = r_2 = -4$.
2. **General Solution for Repeated Roots**:
y(x) = (c_1 + c_2 x)e^{-4x}
3. **Apply Initial Conditions**:
- At $x = 0$: $y(0) = (c_1 + 0)e^0 = c_1 = 1 \implies c_1 = 1$.
- Differentiate to find $y'(x)$:
y'(x) = c_2 e^{-4x} - 4(c_1 + c_2 x)e^{-4x} = [c_2 - 4c_1 - 4c_2 x]e^{-4x}
- At $x = 0$: $y'(0) = c_2 - 4c_1 = 0 \implies c_2 = 4c_1 = 4(1) = 4$.
4. **Exact Particular Solution**:
y(x) = (1 + 4x)e^{-4x}
**3. Comprehensive Options & Distractor Analysis**:
- **(A) y = (1 + 2x)e^(4x)**: ❌ **INCORRECT**. Wrong sign in exponent ($+4x$ instead of $-4x$) and incorrect coefficient $c_2 = 2$.
- **(B) y = (1 - 4x)e^(4x)**: ❌ **INCORRECT**. Positive exponent $+4x$ and negative coefficient $-4$.
- **(C) y = (1 + 8x)e^(-4x)**: ❌ **INCORRECT**. Erroneously takes the linear ODE coefficient (8) as $c_2$.
- **(D) y = (1 + 4x)e^(-4x)**: ✅ **CORRECT**. Perfectly matches the repeated root characteristic solution and initial conditions.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **D**
- **Exam Takeaway**: For repeated roots $r$, the second linearly independent basis solution is $x e^{rx}$. Watch the sign: $r^2 + 8r + 16 = (r+4)^2 = 0 \implies r = -4$.