Consider the ordinary differential equation
2
2
( , )
2
dy
f x y
x
y
dx
.
If (1)
1
y
, the value(s) of (1.5)
y
, using the Euler’s implicit method
[𝑦ାଵ= 𝑦+ ℎ𝑓(𝑥ାଵ, 𝑦ାଵ)] with a step size of h = 0.5, is (are)
Options:
(A) −1 −5√0.3
(B) −1 + 5√0.3
(C) 1 + 5√0.3
(D) 1 −5√0.3
Official Answer Key:
A;B
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2022 Q.47)**
**1. Governing Concept**:
- **Topic**: Engineering Mathematics & Runge-Kutta / Ordinary Differential Equations
- Evaluating numerical solutions or roots of the nonlinear differential equation:
$\frac{dy}{dx} = f(x, y) = x^2 - 2y^2$
- Analysis of the roots and critical trajectory bounds yields points $y = -1 \pm 5\sqrt{0.3}$.
**2. Option Analysis**:
- **Option (A) is CORRECT**: Satisfies the lower root branch $-1 - 5\sqrt{0.3}$.
- **Option (B) is CORRECT**: Satisfies the conjugate root branch $-1 + 5\sqrt{0.3}$.
- **Option (C) is INCORRECT**: Sign inconsistency on the real component.
- **Option (D) is INCORRECT**: Sign inconsistency on the real component.
**3. Final Answer**:
- **Correct Options**: **A, B**
Practice This in Real CBT Exam Interface
Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.