About this topic
Before you begin
Solve first- and second-order linear differential equations, use given substitutions, and analyse exponential and logistic growth with phase lines.
Learning goals
- Solve first-order linear differential equations with an integrating factor
- Reduce a differential equation to a standard linear form with a given substitution
- Interpret solution curves, phase lines and the stability of equilibrium points
- Model exponential and logistic growth, including harvesting
- Solve second-order homogeneous linear differential equations with constant coefficients
- Find a particular integral and the general solution of a non-homogeneous linear equation
Syllabus statements covered
- Include: analytical solution of first order and second order linear differential equations of the form: (i) dy/dx + p(x)y = q(x), using an integrating factor; (ii) d²y/dx² + a dy/dx + by = 0, where a, b ∈ ℝ; (iii) d²y/dx² + a dy/dx + by = f(x), where a, b ∈ ℝ and f(x) is a polynomial or pe^(kx) or p cos(kx) + q sin(kx), including those that can be reduced to the above by means of a given substitution; relationship between the solution of a non-homogenous equation and the associated homogenous equation; family of solution curves; phase lines and slope fields (for teaching and learning only); exponential growth model; logistic growth model, equilibrium points and their stability, and harvesting.
Choose a lesson
Work through them in order.
- Not startedFirst-order equations and growth modelsNextSolve first-order linear equations with an integrating factor, use given substitutions, and analyse growth models with phase lines and harvesting.
- Not startedSecond-order linear differential equationsNextSolve constant-coefficient equations from the auxiliary equation, add a particular integral and fit initial conditions.