Question

Consider the ordinary differential equation $\dfrac{dy}{dx} = f(x,y) = 2x^2 - y^2$.
If $y(1)=1$, find $y(1.5)$ using Euler’s implicit method:
$y_{n+1} = y_n + h f(x_{n+1}, y_{n+1})$ with step size $h = 0.5$.

• A. $-1 - 5\sqrt{0.3}$
• B. $-1 + 5\sqrt{0.3}$
• C. $1 + 5\sqrt{0.3}$
• D. $1 - 5\sqrt{0.3}$

Hint:

Implicit Euler often gives a quadratic equation—always check for two possible roots.

Answer 1

The Correct Option is A, B

Given: $y_0 = y(1) = 1$, step size $h = 0.5$.
Using implicit Euler:
$y_{1} = y_0 + 0.5\,[2(1.5)^2 - y_1^2]$
Compute: $2(1.5)^2 = 4.5$.
Thus, equation becomes:
$y_1 = 1 + 0.5(4.5 - y_1^2)$
$y_1 = 1 + 2.25 - 0.5 y_1^2$
$y_1 = 3.25 - 0.5y_1^2$
Rearranging:
$0.5 y_1^2 + y_1 - 3.25 = 0$
Multiply by 2:
$y_1^2 + 2y_1 - 6.5 = 0$
Using quadratic formula:
$y_1 = \dfrac{-2 \pm \sqrt{4 + 26}}{2} = \dfrac{-2 \pm \sqrt{30}}{2}$
$y_1 = -1 \pm \sqrt{7.5}$
$\sqrt{7.5} = 5\sqrt{0.3}$
So, the two valid roots are:
$y_1 = -1 + 5\sqrt{0.3}$ and $y_1 = -1 - 5\sqrt{0.3}$
Thus, the correct answers are (A) and (B).