Question Type 1: Converting a second order differential equation into a system of first order differential equations Exercises

Use Euler’s method with step size $h=0.1$ on the differential equation $\frac{d^2x}{dt^2}=-2x-\frac{dx}{dt}$, with initial conditions $x(0)=2$ and $x'(0)=0$, to approximate the values of $x(0.3)$ and $x'(0.3)$.

Use Euler's method with a step size of $h = 0.1$ to find the approximate values of $x(0.2)$ and $v(0.2)$.

Apply Euler’s method with step size $h=0.2$ to the second-order differential equation $\frac{d^2x}{dt^2} = e^{-t}x - \frac{dx}{dt} + t$ with initial conditions $x(0)=1$ and $x'(0)=1$. Approximate the values of $x(0.6)$ and $x'(0.6)$.

Apply Euler’s method with step size $h = 0.2$ to the second-order differential equation $\frac{d^2x}{dt^2} = tx - \frac{dx}{dt} + \sin t$ given initial conditions $x(0) = 0$ and $x'(0) = 1$. Find the values of $x(0.4)$ and $x'(0.4)$.

Consider the system of differential equations $\begin{cases} \frac{dx}{dt} = v \\ \frac{dv}{dt} = 2x - v + t^2 \end{cases}$ with initial conditions $x(0) = 4$ and $v(0) = 2$.

Use Euler’s method with step size $h = 0.1$ to find an approximate value for $x(0.5)$ and $v(0.5)$. Give your answers to two decimal places.

Approximate $x(0.4)$ and $x'(0.4)$ for the differential equation $\frac{\text{d}^2x}{\text{d}t^2} = x + t^2$ with $x(0) = 0$ and $x'(0) = 0$, using Euler’s method with a step size of $h = 0.2$.

Consider the second order differential equation $\frac{d^2y}{dt^2} = -3y + 2\frac{dy}{dt} + t$ with initial conditions $y(0) = 1$ and $y'(0) = 0$.

Use Euler’s method with step size $h = 0.1$ to approximate the values of $y(0.3)$ and $y'(0.3)$.

Consider the system of differential equations:

$\frac{\text{d}x}{\text{d}t} = v$ $\frac{\text{d}v}{\text{d}t} = 2x - v + t^2$

with initial conditions $x = 4$ and $v = 2$ at $t = 0$.

Apply Euler’s method with a step size of $h = 0.25$ to approximate the values of $x(1.0)$ and $v(1.0)$. Give your answers to three significant figures.

Use Euler’s method with step size $h = 0.5$ to approximate the values of $x(2.5)$ and $v(2.5)$.

Convert the second order differential equation $\frac{d^2x}{dt^2} = 2x - \frac{dx}{dt} + t^2$ with initial conditions $x(0) = 4$ and $\frac{dx}{dt}(0) = 2$ into an equivalent system of first order differential equations.

Using $h = 0.25$, apply Euler’s method to $\frac{\text{d}^2x}{\text{d}t^2} = \sin t - x$ with $x(0) = 1$ and $x'(0) = 0$ to find the estimates for $x(0.5)$ and $x'(0.5)$.

Using Euler’s method with step size $h=0.1$, compute the approximate values of $x$ and $v = \frac{dx}{dt}$ at $t = 0.1$.