Q.47
Consider the ordinary differential equation
dy/dx = f(x,y) = 2x² – y².

If y(1) = 1, the value(s) of y(1.5), using the Euler’s implicit method
[y_n+1 = y_n + h f(x_n+1, y_n+1)] with a step size of h = 0.5, is (are)

(A)	−1 − 5√0.3
(B)	−1 + 5√0.3
(C)	1 + 5√0.3
(D)	1 − 5√0.3

 

Implicit Euler Method Example – Solve dy/dx = 2x² – y² |

Question

Consider the differential equation:

dy/dx = 2x² − y²

Given y(1) = 1, compute y(1.5) using the Euler implicit method:

y_n+1 = y_n + h f(x_n+1, y_n+1)

Step size: h = 0.5

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Step-by-Step Solution

Starting point:

• x₀ = 1
• y₀ = 1
• One step of size h = 0.5 → x₁ = 1.5

Apply implicit Euler:

y₁ = y₀ + 0.5(2(1.5)² − y₁²)

y₁ = 1 + 0.5(4.5 − y₁²)

y₁ = 3.25 − 0.5y₁²

Rewriting:

0.5y₁² + y₁ − 3.25 = 0

Multiply by 2:

y₁² + 2y₁ − 6.5 = 0

Using the quadratic formula:

y₁ = [-2 ± √(4 + 26)] / 2

y₁ = [-2 ± √30] / 2

Since √30 = 5√0.3:

y₁ = -1 ± 5√0.3

Final Answer

Correct options:

• (A) −1 − 5√0.3
• (B) −1 + 5√0.3

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Explanation of Options

Option	Expression	Correct?	Reason
A	−1 − 5√0.3	✔ Yes	Valid root of quadratic
B	−1 + 5√0.3	✔ Yes	Valid root of quadratic
C	1 + 5√0.3	✘ No	Wrong sign
D	1 − 5√0.3	✘ No	Does not satisfy equation

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