GATE BT 2024 • Q30
MCQ • 1 Mark
Section 1: Engineering Mathematics
Differential Equations
Subtopic: Fundamentals of Differential Equations
Question Statement:
The solution of the differential equation
dy
dx = y + e-x that satisfies y(0) = -
1
2 is
________.
Options:
-
(A) -
1
2 e-x
2
-
(B) -
1
2 ex
-
(C) -
1
2 e-x
-
(D) -
1
2 e
x
2
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2024 Q.30)**
**1. Governing Concept**:
- **Topic**: Differential Equations (First-Order Linear ODE)
- Standard form:
$\frac{dy}{dx} - y = e^{-x}$
Here $P(x) = -1$ and $Q(x) = e^{-x}$.
- Integrating factor ($I.F.$):
$I.F. = e^{\int -1 \, dx} = e^{-x}$
- General solution:
$y \cdot e^{-x} = \int e^{-x} \cdot e^{-x} \, dx = \int e^{-2x} \, dx = -\frac{1}{2} e^{-2x} + C$
$y = -\frac{1}{2} e^{-x} + C e^x$
- Applying initial condition $y(0) = -\frac{1}{2}$:
$-\frac{1}{2} = -\frac{1}{2} e^0 + C e^0 \implies -\frac{1}{2} = -\frac{1}{2} + C \implies C = 0$
- Particular solution:
$y = -\frac{1}{2} e^{-x}$
**2. Option Analysis**:
- **Option (A) is INCORRECT**: Erroneous exponent $-x/2$.
- **Option (B) is INCORRECT**: Wrong sign in exponent ($e^x$).
- **Option (C) [-1/2 e^(-x)] is CORRECT**: Directly matches $y(x) = -\frac{1}{2} e^{-x}$.
- **Option (D) is INCORRECT**: Exponent error.
**3. Final Answer**:
- **Correct Choice**: **C**
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