Differential Equations

### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2026 Q.58)** **1. Governing Concept & Definition**: - **Topic**: Differential Equations & Fundamental Theorem of Calculus - The definite integral of a derivative represents net change: $\int_2^3 \frac{dy}{dx}\,dx = y(3) - y(2) \implies y(3) = y(2) + \int_2^3 \frac{dy}{dx}\,dx$ - The integral corresponds to the area under the curve $\frac{dy}{dx}$ from $x = 2$ to $x = 3$. **2. Detailed Step-by-Step Calculation**: - **From Plot**: - At $x = 2$, $\frac{dy}{dx} = 3$ - At $x = 3$, $\frac{dy}{dx} = 4$ - The region between $x=2$ and $x=3$ is a linear trapezoid with heights 3 and 4, and width $\Delta x = 1$. - **Step 1: Calculate the area under the derivative curve**: $\text{Area} = \frac{3 + 4}{2} \times (3 - 2) = \frac{7}{2} \times 1 = 3.5$ - **Step 2: Compute $y(3)$**: $y(3) = y(2) + 3.5 = 5 + 3.5 = 8.5$ **3. Final Answer**: - **Calculated Value**: **8.5** - **Official GATE Accepted Range**: **8.3 to 8.7**