IB Mathematics Applications & Interpretation 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 $\frac{d^{2}x}{d{t}^2}=-2x-\frac{dx}{dt}$, with initial conditions $x(0)=2$ and $x^{\prime }(0)=0$, to approximate the values of $x(0.3)$ and $x^{\prime }(0.3)$.

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 $$\frac{d^{2}x}{d{t}^2}=e^{-t}x-\frac{dx}{dt}+t$$ with initial conditions $x(0)=1$ and $x^{\prime }(0)=1$. Approximate the values of $x(0.6)$ and $x^{\prime }(0.6)$.

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

Consider the system of differential equations $$\{\begin{array}{l}\frac{dx}{dt}=v\\ \frac{dv}{dt}=2x-v+t^2\end{array}$$ 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 $x^{\prime }(0.4)$ for the differential equation $$\frac{{\text{d}}^{2}x}{\text{d}{t}^2}=x+t^2$$ with $x(0)=0$ and $x^{\prime }(0)=0$, using Euler’s method with a step size of $h=0.2$.

Consider the second order differential equation $$\frac{d^{2}y}{d{t}^2}=-3y+2\frac{dy}{dt}+t$$ with initial conditions $y(0)=1$ and $y^{\prime }(0)=0$.

Use Euler’s method with step size $h=0.1$ to approximate the values of $y(0.3)$ and $y^{\prime }(0.3)$.

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 $$\frac{d^{2}x}{d{t}^2}=2x-\frac{dx}{dt}+t^2$$ 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 $\frac{{\text{d}}^{2}x}{\text{d}{t}^2}=\sin t-x$ with $x(0)=1$ and $x^{\prime }(0)=0$ to find the estimates for $x(0.5)$ and $x^{\prime }(0.5)$.

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