Exercises for Question Type 2: Applying Euler's method to second order differential equations - IB | RevisionDojo

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

Question 1

Skill question

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 $t=0.3$ .

Question 2

Skill question

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 $\dot x(0)=2$ .

Question 3

Skill question

Derive the general Euler iteration formulas for a second order ODE

\frac{d^2x}{dt^2}=f(t,x,\dot x)

by converting it into a system of first order equations. Then specify these formulas for $f(t,x,y)=2x-y+t^2$ .

Question 4

Skill question

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 $\dot x(0)=2$ .

Question 5

Skill question

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 $\dot x(0)=2$ .

Question 6

Skill question

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.

Question 7

Skill question

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 $\dot x(0)=2$ .

Question 8

Skill question

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 $\dot x(0)=2$ .

Question 9

Skill question

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 $\dot x(0)=2$ .

Question 10

Skill question

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 behaviour of the approximate solution as $t$ increases.

Question 11

Skill question

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 $t_n$ , $x_n$ , $\dot x_n$ .

Question 12

Skill question

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

given $x(0)=4$ and $\dot x(0)=2$ . Discuss how the step size affects the numerical accuracy.

Question 1

Skill question

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 $t=0.3$ .

Question 2

Skill question

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 $\dot x(0)=2$ .

Question 3

Skill question

Derive the general Euler iteration formulas for a second order ODE

\frac{d^2x}{dt^2}=f(t,x,\dot x)

by converting it into a system of first order equations. Then specify these formulas for $f(t,x,y)=2x-y+t^2$ .

Question 4

Skill question

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 $\dot x(0)=2$ .

Question 5

Skill question

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 $\dot x(0)=2$ .

Question 6

Skill question

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.

Question 7

Skill question

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 $\dot x(0)=2$ .

Question 8

Skill question

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 $\dot x(0)=2$ .

Question 9

Skill question

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 $\dot x(0)=2$ .

Question 10

Skill question

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 behaviour of the approximate solution as $t$ increases.

Question 11

Skill question

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 $t_n$ , $x_n$ , $\dot x_n$ .

Question 12

Skill question

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

given $x(0)=4$ and $\dot x(0)=2$ . Discuss how the step size affects the numerical accuracy.