GATE BT 2026 • Q27
MSQ • 1 Mark
Section 1: Engineering Mathematics
Differential Equations
Subtopic: Fundamentals of Differential Equations
Question Statement:
The equation
d2y
dx2 - y = 0 has a solution of the form y = e Ax. The value(s) of
A satisfying this is/are:
Options:
-
(A) 0
-
(B) 1
-
(C) -1
-
(D) -∞
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2026 Q.27)**
**1. Governing Concept**:
- **Topic**: Engineering Mathematics & Second-Order Ordinary Differential Equations (ODEs)
- Given homogeneous linear ODE with constant coefficients:
$\frac{d^2y}{dx^2} - y = 0$
- Substituting trial solution $y = e^{Ax}$:
$\frac{dy}{dx} = A e^{Ax}, \quad \frac{d^2y}{dx^2} = A^2 e^{Ax}$
$A^2 e^{Ax} - e^{Ax} = 0 \implies (A^2 - 1) e^{Ax} = 0$
- Since $e^{Ax} \neq 0$ for all real $x$, the characteristic equation is:
$A^2 - 1 = 0 \implies A^2 = 1 \implies A = \pm 1$
**2. Option Analysis**:
- **Option (A) [0] is INCORRECT**: Substituting $A = 0$ gives $y = e^0 = 1 \implies y'' = 0 \implies 0 - 1 = -1 \neq 0$.
- **Option (B) [1] is CORRECT**: $A = 1$ gives $y = e^x$, where $\frac{d^2(e^x)}{dx^2} - e^x = e^x - e^x = 0$.
- **Option (C) [-1] is CORRECT**: $A = -1$ gives $y = e^{-x}$, where $\frac{d^2(e^{-x})}{dx^2} - e^{-x} = e^{-x} - e^{-x} = 0$.
- **Option (D) [-∞] is INCORRECT**: $-\infty$ is not a finite scalar solution.
**3. Final Answer**:
- **Correct Options**: **B, C**
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