IB Mathematics Applications & Interpretation Question Type 2: Applying Euler's Method to Second Order Differential Equations Exercises

Apply Euler's method with $h=0.5$ to approximate $x(t)$ and $\dot{x}(t)$ at $t=2.5$ for the differential equation $$\frac{d^{2}x}{d{t}^2}=2x-\frac{dx}{dt}+t^2$$ given $x(0)=4$ and $\dot{x}(0)=2$. Present your results in a table with columns $t_n$, $x_n$, ${\dot{x}}_n$.

Use Euler's method with step length $h=0.5$ to approximate $x(2.5)$ and $\dot{x}(2.5)$ for the differential equation $$\frac{d^{2}x}{d{t}^2}=2x-\frac{dx}{dt}+t^2$$ given $x(0)=4$ and $\dot{x}(0)=2$.

Use Euler's method with step length $h=0.3$ to approximate $x(1.5)$ and $\dot{x}(1.5)$ for the differential equation $$\frac{d^{2}x}{d{t}^2}=2x-\frac{dx}{dt}+t^2$$ given $x(0)=4$ and $\dot{x}(0)=2$.

Use Euler's method with step length $h=0.2$ to approximate $x(1.0)$ and $\dot{x}(1.0)$ for the differential equation $$\frac{d^{2}x}{d{t}^2}=2x-\frac{dx}{dt}+t^2$$ given $x(0)=4$ and $\dot{x}(0)=2$. Then comment on the qualitative behavior of the approximate solution as $t$ increases.

Derive the general Euler iteration formulas for a second order ODE $$\frac{{\text{d}}^{2}x}{\text{d}{t}^2}=f(t,x,\dot{x})$$ by converting it into a system of first order equations. Then specify these formulas for $f(t,x,y)=2x-y+t^2$.

Use Euler's method with step length $h=0.1$ to approximate $x(t)$ and $\dot{x}(t)$ for the differential equation $$\frac{d^{2}x}{d{t}^2}=2x-\frac{dx}{dt}+t^2$$ given $x(0)=4$, $\dot{x}(0)=2$. Show only the first three iterations (up to $t=0.3$) and state your approximate values at $t=0.3$.

Use Euler's method with step length $h=0.4$ to approximate $x(2.0)$ and $\dot{x}(2.0)$ for the differential equation $$\frac{d^{2}x}{d{t}^2}=2x-\frac{dx}{dt}+t^2$$ given $x(0)=4$ and $\dot{x}(0)=2$.

Use Euler's method with step length $h=0.1$ to approximate $x(0.5)$ and $\dot{x}(0.5)$ for the differential equation $$\frac{d^{2}x}{d{t}^2}=2x-\frac{dx}{dt}+t^2$$ given $x(0)=4$ and $\dot{x}(0)=2$.

Compare the approximations of $x(0.5)$ and $\dot{x}(0.5)$ obtained by Euler's method with step sizes $h=0.1$ and $h=0.25$ for the differential equation $$\frac{{\text{d}}^{2}x}{\text{d}{t}^2}=2x-\frac{\text{d}x}{\text{d}t}+t^2$$ given $x(0)=4$ and $\dot{x}(0)=2$. Discuss how the step size affects the numerical accuracy.

Use Euler's method with step length $h=0.2$ to approximate $x(1.0)$ and $\dot{x}(1.0)$ for the differential equation $$\frac{d^{2}x}{d{t}^2}=2x-\frac{dx}{dt}+t^2$$ given $x(0)=4$ and $\dot{x}(0)=2$.

Use Euler's method with step length $h=0.25$ to approximate $x(1.25)$ and $\dot{x}(1.25)$ for the differential equation $$\frac{d^{2}x}{d{t}^2}=2x-\frac{dx}{dt}+t^2$$ given $x(0)=4$ and $\dot{x}(0)=2$.

Outline an algorithm to approximate $x(2.5)$ and $\dot{x}(2.5)$ using Euler's method with $h=0.1$ for the equation $$\frac{d^{2}x}{d{t}^2}=2x-\frac{dx}{dt}+t^2$$ given $x(0)=4$, $\dot{x}(0)=2$. State how many steps are required and how you would implement the iteration in a spreadsheet or program.