Reducible Differential Equations

• Reducible differential equations are a type of differential equations that can be rewritten or simplified to standard forms which are already known and whose solutions are already established.

• They usually involve a substitution of variables which reduces the equation to a simpler or more manageable form. The purpose of reduction is to make the methods for solving the original problem easier.

• To recognise a reducible differential equation, look for expressions of a function and its derivatives multiplied together that can be rewritten as the derivative of a new function. If such a derivative can be identified, the substitution can be made.

• A common example of a reducible differential equation is the homogeneous differential equation. By a change of variables, this can be reduced to a variable separable equation. This process often involves introducing a new variable v such that y = vx.

• Another type of reducible differential equation is a linear differential equation. By using an integrating factor, this can be transformed to an equation that is easily integrable.

• It’s important to mention the Bernoulli differential equation. It is a nonlinear differential equation of the form y’ + p(x)y = q(x)y^n. Despite it’s nonlinearity, it can be reduced to a linear differential equation via a simple substitution.

• Always remember to substitute the original variables back in, and check your solution against the original differential equation to ensure it’s correct.

• Practice problems! The best way to familiarise yourself with the process of recognising and reducing differential equations is to solve lots of problems. Don’t just rely on recognising the forms, understand why the methods work.

• Lastly, when faced with a differential equation, first identity which type of equation you are dealing with. Sometimes the reduction can be nontrivial, so keep an eye on possible substitutions or transformations.

Note: Always remember to verify your solutions!