PYQs Based Test - 28 : Numerical Methods (Differential Equations)

Consider the first order initial value problem

y' = y + 2x − x², y(0) = 1, (0 ≤ x < ∞)

with exact solution y(x) = x² + e${}^x$. For x = 0.1, the percentage difference between the exact solution and the solution obtained using a single iteration of the second-order Runge – Kutta method with step-size h = 0.1 is __________

The ordinary differential equation $\frac{dy}{dt}=-\pi y$ subject to an initial condition y(0) = 1 is solved numerically using the following scheme:

$\frac{y(t_{n+1})-y(t_n)}{h}=-\pi y(t_n)$

where h is the time step, tₙ = nh, and n = 0, 1, 2, .... This numerical scheme is stable for all values of h in the interval ______.

The ordinary differential equation $\frac{\mathrm{dy}}{\mathrm{dt}}=-3\mathrm{x}+2,\mathrm{with}\mathrm{x}\left(0\right)=1$

is to be solved using the forward Euler method. The largest time step that can be used to solve the equation without making the numerical solution unstable is ________.

Runge-Kutta fourth order method is used to solve the differential equation $\frac{dy}{dx}=(y-x)$ . If the initial value y(0) = 2 and step-size is 0.1, then the value of y(0.1) is ______ (up to three decimal places)

If, $\frac{dy}{dx}=x+y,y\left(0\right)=1$ using Runge’s method the value of y at x = 0.2, when h = 0.2 is