Question

The differential equation $\frac{dy}{dx}=0.75y^2$ is to be solved using the backward (implicit) Euler's method with the boundary condition y = 1 at x = 0 and with a step size of 1. What would be the value of y at x = 1?

• A. $\frac{3}{2}(\sqrt{2}\pm i)$
• B. $\frac{3}{2}\left(1\pm \sqrt{2}\right)$
• C. $\frac{2}{3}\left(1\pm i\sqrt{2}\right)$
• D. $\frac{3}{2}(2\pm i)$

Answer 1

The correct option is C $\frac{2}{3}\left(1\pm i\sqrt{2}\right)$

$\frac{dy}{dx}=0.75y^2(y=1atx=0)$

Iterative equation by backward (implicit) Euler's method for above equation would be

$y_{k+1}=y_k+h_f(x_{k+1},y_{k+1})$

$y_{k+1}=y_k+h\times 0.75y_{k+1}^2$

$\Rightarrow 0.75h{y}_{k+1}^2-y_{k+1}+y_k=0$

Putting k = 0 in above equation
$0.75h{y}_1^2-y_1+y_0=0$
Since $y_0$ = 1 and h = 1
$0.75y_1^2-y_1+1=0$
$\Rightarrow y_1=\frac{1\pm \sqrt{1^2-3}}{2\times 0.75}=\frac{2}{3}(1\pm i\sqrt{2})$