First Order Differential Equations - Still to be done

First Order Differential Equations - Still to be done

First Order Differential Equations

Common Question Formats

• Solving a first order differential equation with given initial conditions.
• Finding the particular solution of a first order differential equation based on given information.
• Verifying solutions of a first order differential equation.
• Creating a model of a real-world phenomenon using a first order differential equation.
• Analysis of the behaviour of functions modelled by first order differential equations.

Strategies for Addressing Questions

• Identify if the differential equation is linear or separable. This will guide the appropriate method to use in solving.
• For linear equations, rearrange into the standard form, i.e., in the form (\frac{dy}{dx} + p(x)y = q(x)), then apply an integrating factor to solve.
• For separable equations, rearrange the equation such that variables of the same kind are together (all x’s on one side and all y’s on the other side). Integrate both sides separately.
• Always include the arbitrary constant, (+C), when integrating.
• If given an initial condition, substitute it into the general solution to find the specific solution.
• If asked to create a model, use the variables and the relationships given to form a differential equation. Proceed with solving as usual.

Potential Pitfalls

• Forgetting to include the arbitrary constant (+C) when integrating, which can lead to an incomplete solution.
• Misidentifying the form of the differential equation. This can result in using an inappropriate method for solving.
• Neglecting to simplify where possible. Oftentimes, simplifying can make the integration process easier.
• Assuming that a differential equation can only model a physical process. They also model biological, chemical, and economic phenomena, among others.
• Forgetting to verify the solution. Substituting the solution back into the original equation can detect calculation errors.