Subtopic: Higher Order Linear ODEs with Constant Coefficients
Question Statement:
Consider the differential equation with the initial conditions given below. If y(x)
is the solution of the equation, the value of the slope,
$\frac{dy}{dx}$ , at x = ln(2) is
________ (rounded off to three decimal places).
$\frac{d^2y}{dx^2}$ + 2 $\frac{dy}{dx}$ + y = 0 with y|x=0 = 0 and $\frac{dy}{dx}$|
x=0
= 1
Official Answer Key:
0.153
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2026 Q.49)**
**1. Intuitive Physical Setup & Given Parameters**:
We are given the second-order linear homogeneous ordinary differential equation:
\frac{d^2y}{dx^2} + 2\frac{dy}{dx} + y = 0
with initial conditions:
y(0) = 0, \quad \left.\frac{dy}{dx}\right|_{x=0} = 1
We need to find the value of the slope $\frac{dy}{dx}$ at $x = \ln(2)$, rounded off to 3 decimal places.
**2. Step-by-Step Mathematical Derivation**:
1. **Characteristic Equation**:
r^2 + 2r + 1 = 0 \implies (r + 1)^2 = 0 \implies r = -1 \text{ (repeated root)}
2. **General Solution**:
y(x) = (C_1 + C_2 x) e^{-x}
3. **Apply Initial Conditions**:
- At $x = 0$:
y(0) = C_1 = 0
Thus, $y(x) = C_2 x e^{-x}$.
- First derivative:
\frac{dy}{dx} = C_2 \frac{d}{dx}(x e^{-x}) = C_2 (1 - x) e^{-x}
- At $x = 0$:
\left.\frac{dy}{dx}\right|_{x=0} = C_2 (1 - 0) e^0 = C_2 = 1
- The unique solution and its derivative are:
y(x) = x e^{-x}
\frac{dy}{dx} = (1 - x) e^{-x}
4. **Evaluate Slope at $x = \ln(2)$**:
- Note that $e^{-\ln(2)} = \frac{1}{e^{\ln(2)}} = \frac{1}{2} = 0.5$.
- $\ln(2) \approx 0.693147$.
- Substitute into the slope expression:
\left.\frac{dy}{dx}\right|_{x=\ln(2)} = (1 - \ln 2) e^{-\ln 2} = \frac{1 - 0.693147}{2} = \frac{0.306853}{2} \approx 0.153426
**3. Comprehensive Options & Distractor Analysis**:
- **Evaluation Criteria**: Numerical Answer Type (NAT).
- **Exact Analytical Result**: $\frac{1 - \ln 2}{2} \approx 0.1534$.
- **Rounding Requirement**: Three decimal places $\implies \mathbf{0.153}$.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **0.153** (Acceptable Range: **0.150 to 0.156**)
- **Exam Takeaway**: For critically damped ODEs with repeated root $r$, the general solution takes the form $(C_1 + C_2 x)e^{rx}$. Evaluate derivatives carefully using the product rule.