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 $y^{\mathrm{\prime }}=x(y+x)-1$, $y(0)=2$.

[2013, 15M]

2) Provide a computer algorithm to solve an ordinary differential equation ${\displaystyle \frac{dy}{dx}}=f(x,y)$ 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 ${\displaystyle \frac{dy}{dx}}$ at $x=0.1$ from the following data:
$\begin{array}{|ccccc|}\hline x& 0.1& 0.2& 0.3& 0.4\\ y& 0.9975& 0.9900& 0.9776& 0.9604\\ \hline\end{array}$

[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 ${\displaystyle \frac{dy}{dx}}={\displaystyle \frac{y^2-x^2}{y^2+x^2}}$ 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 ${\displaystyle \frac{dy}{dx}}=x(y-x)$, $y(2)=3$ 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 ${\displaystyle \frac{dy}{dx}}=\sqrt{x+y}$, $y(0.4)=0.41$. 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: $y^{\mathrm{\prime }}=xy$, $y(1)=2$.

[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 $y^{\mathrm{\prime }}=x+y$, $y(0)=1$.

[2007, 15M]

6) Given ${\displaystyle \frac{dy}{dx}}=y-x$ 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]