Differential Equations

### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2017 Q.16)** **1. Governing Concept & Definition**: - **Topic**: Engineering Mathematics & Differential Equations - Given $\frac{d^2 y}{dx^2} = 0$, the general solution is linear: $y(x) = C_1 x + C_2$ **2. Detailed Step-by-Step Calculation**: - Derivative: $\frac{dy}{dx} = C_1$. - Boundary condition 1: $\frac{dy}{dx} = 0$ at $x = 0 \implies C_1 = 0$. So $y(x) = C_2$ (a constant function). - Boundary condition 2: $y(1) = 1 \implies C_2 = 1$. - Therefore, $y(x) = 1$ for all $x$. - At $x = 2$: $y(2) = 1$ **3. Final Answer**: - **Calculated Value**: **1** - **Official GATE Accepted Range**: **1 to 1**