Modelling with Differential Equations
First Order Differential Equations
Second and Higher Order Differential Equations
Simultaneous Differential Equations
Numerical Methods
Construction of models
Interpretation of solutions
Tangent Fields
Equations with separable variables
First order linear differential equations
Homogeneous second order differential equations
The general second order differential equation
Damped oscillations
Higher order linear differential equations
Simultaneous linear differential equations
Step by step methods
- Understand how to introduce and define variables to describe a given situation in mathematical terms.
- Be able to relate 1st and 2nd order derivatives to verbal descriptions and so formulate differential equations.
- Know the language of kinematics, and the relationships between the various terms.
- Know Newton's 2nd law of motion
- Understand how to determine the order of a differential equation
- Be able to interpret the solution of a differential equation in terms of the original situation
- Appreciate the difference between a general solution and a particular solution, i.e. one which satisfies particular prescribed conditions
- Understand the significance of the number of arbitrary constants in a general solution
- Be able to investigate the effect of changing a differential equation on its solution
- Be able to sketch the tangent field for a 1st order differential equation and be able to interpret it
- Be able to sketch and interpret the curve of the solution corresponding to particular conditions
- Be able to identify isoclines and use them in sketching and interpreting tangent fields
- Be able to find both general and particular solutions of a 1st order differential equation with separable variables
- Be able to solve 1st order differential equations with constant coefficients
- Be able to distinguish differential equations where the integrating factor method is appropriate, and to rearrange such equations if necessary
- Be able to find an integrating factor and understand its significance in the solution in the solution of an equation
- Be able to solve an equation using an integrating factor and find both general and particular solutions
- Be able to solve homogeneous 2nd order differential equations, using the auxiliary equation and complementary function
- Appreciate the relationship between different cases of the solution and the nature of the roots of the auxiliary equation, and be able to interpret these different cases graphically
- Be able to find the particular solution in given contexts
- Be able to solve the general 2nd order linear differential equation, by solving the homogeneous case and adding a particular integral.
- Be able to find particular integrals in simple cases.
Appreciate the relationship between different cases of the solution and the nature of the roots of the auxiliary equation, and be able to interpret these different cases graphically
- Be able to solve the equation for simple harmonic motion, x+ω(x+k)=0, and be able to relate the various forms of the solution to each other
- Be able to model damped oscillations using 2nd order linear differential equations, and understand the associated terminology
- Be able to interpret the solutions of equations modelling damped oscillations in words and graphically
- Appreciate that the same methods can be extended to higher order equations and be able to do so in simple cases
- Model situations with one independent variable and two dependent variables which lead to 1st order simultaneous differential equations, and know how to solve these by eliminating one variable to produce a single, 2nd order equation
- Appreciate that the same method can be extended to more than two such equations, leading by elimination to a single higher order equation
- Be able to use step by step methods (e.g. Euler's method) to solve 1st order differential equations (including simultaneous equations) where appropriate