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Second Order Differential Equations (DP IB Maths: AI HL)
Revision Note
Euler's Method: Second Order
How do I apply Euler’s method to second order differential equations?
- A second order differential equation is a differential equation containing one or more second derivatives
- In this section of the course we consider second order differential equations of the form
- You may need to rearrange the differential equation given to get it in this form
- In order to apply Euler’s method, use the substitution to turn the second order differential equation into a pair of coupled first order differential equations
- If , then
- This changes the second order differential equation into the coupled system
- Approximate solutions to this coupled system can then be found using the standard Euler’s method for coupled systems
- See the notes on this method in the revision note 5.6.4 Approximate Solutions to Differential Equations
Worked example
Consider the second order differential equation .
a)
Show that the equation above can be rewritten as a system of coupled first order differential equations.
b)
Initially and . By applying Euler’s method with a step size of 0.1, find approximations for the values of x and when t = 0.5 .
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Exact Solutions & Phase Portraits: Second Order
How can I find the exact solution for a second order differential equation?
- In some cases we can apply methods we already know to find the exact solutions for second order differential equations
- In this section of the course we consider second order differential equations of the form
-
- are constants
- Use the substitution to turn the second order differential equation into a pair of coupled first order differential equations
- If , then
- This changes the second order differential equation into the coupled system
-
- The coupled system may also be represented in matrix form as
- In the ‘dot notation’ here and
- That can be written even more succinctly as
- Here , , and
- Once the original equation has been rewritten in matrix form, the standard method for finding exact solutions of systems of coupled differential equations may be used
- The solutions will depend on the eigenvalues and eigenvectors of the matrix M
- For the details of the solution method see the revision note 5.7.1 Coupled Differential Equations
- Remember that exam questions will only ask for exact solutions for cases where the eigenvalues of M are real and distinct
How can I use phase portraits to investigate the solutions to second order differential equations?
- Here we are again considering second order differential equations of the form
- a & b are real constants
- As shown above, the substitution can be used to convert this second order differential equation into a system of coupled first order differential equations of the form
- Here , , and
- Once the equation has been rewritten in this form, you may use the standard methods to construct a phase portrait or sketch a solution trajectory for the equation
- For the details of the phase portrait and solution trajectory methods see the revision note 5.7.1 Coupled Differential Equations
- When interpreting a phase portrait or solution trajectory sketch, don’t forget that
- So if x represents the displacement of a particle, for example, then will represent the particle’s velocity
Worked example
Consider the second order differential equation . Initially x = 3 and .
a)
Show that the equation above can be rewritten as a system of coupled first order differential equations.
b)
Given that the matrix has eigenvalues of 1 and -4 with corresponding eigenvectors and , find the exact solution to the second order differential equation.
c)
Sketch the trajectory of the solution to the equation on a phase diagram, showing the relationship between x and .
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