PYQs

1) Use Euler’s method with step size \(h=0.15\) to compute the approximate value of \(y(0.6)\), correct up to five decimal places from the initial value problem, where \(y^{\prime}=x(y+x)-1\), \(y(0)=2\).

[2013, 15M]

2) Provide a computer algorithm to solve an ordinary differential equation \(\dfrac{d y}{d x}=f(x, y)\) in the interval \([a, b]\) for \(n\) number of discrete points, where the initial value is \(y(a)=\alpha\), using Euler’s method.

[2012, 12M]

3) Find \(\dfrac{d y}{d x}\) at \(x=0.1\) from the following data:
\(\begin{array}{|c|c|c|c|c|c|}\hline {x} & {0.1} & {0.2} & {0.3} & {0.4}\\ \hline {y} & {0.9975} & {0.9900} & {0.9776} & {0.9604}\\ \hline\end{array}\)

[2012, 20M]

4) Suppose a computer spends 60 per cent of its time handling a particular type of computation when running a given program and its manufacturers make a change that improves its performance on that type of computation by a factor of 10. If the program takes 100 sec to execute, what will its execution time be after the change?

[2010, 6M]

1) Using Runge-Kutta method of fourth order, solve \(\dfrac{dy}{dx}=\dfrac{y^2-x^2}{y^2+x^2}\) with \(y(0)=1\) at \(x=0.2\). Use four decimal places for calculation and step length 0.2.

[2019, 10M]

2) Solve the initial value problem \(\dfrac{d y}{d x}=x(y-x)\), \(y(2)=3\) in the interval \([2,2.4]\) using the Runge-Kutta fourth-order method with step size \(h=0.2\).

[2015, 15M]

3) Use Runge-Kutta formula of fourth order to find the value of \(y\) at \(x=0.8\), where \(\dfrac{d y}{d x}=\sqrt{x+y}\), \(y(0.4)=0.41\). Take the step length \(h=0.2\).

[2014, 20M]

4) Find the value of \(y(1.2)\) using Runge-Kutta fourth order method with step size \(h=0.2\) from the initial value problem: \(y^{\prime}=x y\), \(y(1)=2\).

[2009, 15M]

5) Apply the second order Runge-Kutta method to find an approximate value of \(y\) at \(x=0.2\) taking \(h=0.1\), given that \(y\) satisfies the differential equation and the initial condition \(y^{\prime}=x+y\), \(y(0)=1\).

[2007, 15M]

6) Given \(\dfrac{d y}{d x}=y-x\) where \(y(0)=2\), using the Runge-Kutta fourth order method, find \(y(0.1)\) and \(y(0.2)\). Compare the approximate solution with its exact solution. \((e^{0.1}=1.10517\), \(e^{0.2}=1.2214)\)

[2002, 20M]