Teaching CSIR NET Mock Test Series Mathematical Science Ordinary Differential Equations Initial Value Problem
Consider the initial value problem
\(\frac{d y}{d x}=f(x, y)\), y(x0) = y0
where f is a twice continuously differentiable function on a rectangle containing the point (x0, y0). With the step-size h, let the first iterate of a second order scheme to approximate the solution of the above initial value problem be given by
y1 = y0 + Pk1 + Qk2,
where k1 = hf(x0, y0), k2 = hf(x0 + α0h, y0 + β0k1) and P, Q, α0, β0 ∈ ℝ.
Which of the following statements are correct?
1
If α0 = 2, then β0 = 2, \(P=\frac{3}{4}, Q=\frac{1}{4}\)
2
If β0 = 3, then α0 = 3, \(P=\frac{5}{6}, Q=\frac{1}{6}\)
3
If α0 = 2, then β0 = 2, \(P=\frac{1}{4}, Q=\frac{3}{4}\)
4
If β0 = 3, then α0 = 3, \(P=\frac{1}{6}, Q=\frac{5}{6}\)