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GATE 2022 BT – Question 47

Engineering Mathematics · Numerical Methods: Linear and nonlinear algebraic equations, trapezoidal and Simpson's rule, single step methods for ODEs · 2 marks · Multiple select

For $dy/dx=2x^2-y^2$, $y(1)=1$. With implicit Euler $y_{n+1}=y_n+hf(x_{n+1},y_{n+1})$ and h=0.5, the value(s) of y(1.5) is (are)

  1. $-1-5\sqrt{0.3}$
  2. $-1+5\sqrt{0.3}$
  3. $1+5\sqrt{0.3}$
  4. $1-5\sqrt{0.3}$

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Correct answer: (A) $-1-5\sqrt{0.3}$; (B) $-1+5\sqrt{0.3}$

Explanation

**Implicit (backward) Euler method:**
$$y_{n+1}=y_n+h\,f(x_{n+1},\,y_{n+1}),\qquad f(x,y)=2x^2-y^2 .$$

**Numbers.** $x_0=1$, $y_0=1$, $h=0.5$, so $x_1=1.5$:
$$y=1+0.5\left(2(1.5)^2-y^2\right)=1+0.5(4.5-y^2)=3.25-0.5y^2 .$$

**Rearrange into a quadratic:**
$$0.5y^2+y-3.25=0\;\Rightarrow\;y^2+2y-6.5=0\;\Rightarrow\;(y+1)^2=7.5 .$$

$$y=-1\pm\sqrt{7.5}=-1\pm5\sqrt{0.3}\quad(\text{since }\sqrt{7.5}=5\sqrt{0.3}).$$

Because the update is implicit, the algebraic equation has two roots, and both are possible values: $-1-5\sqrt{0.3}$ and $-1+5\sqrt{0.3}$ (options **A and B**).