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Question 1

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^2x}{dt^2} = 2x - \frac{dx}{dt} + t^2

given $x(0) = 4$ and $\dot{x}(0) = 2$ . Present your results in a table with columns $t_n$ , $x_n$ , $\dot{x}_n$ .

[5]

Question 2

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^2x}{dt^2}=2x-\frac{dx}{dt}+t^2

given $x(0)=4$ and $\dot x(0)=2$ .

[6]

Question 3

This question assesses the application of Euler's method to approximate solutions for a second-order differential equation by transforming it into a system of two first-order differential equations.

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^2x}{dt^2}=2x-\frac{dx}{dt}+t^2

given $x(0)=4$ and $\dot x(0)=2$ .

[6]

Question 4

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^2x}{dt^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.

[7]

Question 5

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

[5]

Question 6

Use Euler's method with step length $h=0.1$ to approximate $x(t)$ and $\dot x(t)$ for the differential equation

\frac{d^2x}{dt^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 $t=0.3$ .

[5]

Question 7

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^2x}{dt^2} = 2x - \frac{dx}{dt} + t^2

given $x(0) = 4$ and $\dot x(0) = 2$ .

[6]

Question 8

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^2x}{dt^2}=2x-\frac{dx}{dt}+t^2

given $x(0)=4$ and $\dot x(0)=2$ .

[6]

Question 9

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 $\frac{\text{d}^2x}{\text{d}t^2}=2x-\frac{\text{d}x}{\text{d}t}+t^2$ given $x(0)=4$ and $\dot x(0)=2$ . Discuss how the step size affects the numerical accuracy.

[7]

Question 10

Mathematic HL/Further Mathematics

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^2x}{dt^2} = 2x - \frac{dx}{dt} + t^2

given $x(0) = 4$ and $\dot{x}(0) = 2$ .

[6]

Question 11

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^2x}{dt^2}=2x-\frac{dx}{dt}+t^2

given $x(0)=4$ and $\dot{x}(0)=2$ .

[6]

Question 12

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^2x}{dt^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.

[6]

Question Type 2: Applying Euler's method to second order differential equations Exercises