First order ODEs

First-order linear differential equation

A first-order, linear ODE is given the standard form¹

𝑑𝑦𝑑𝑥+𝑝(𝑥)𝑦=𝑔(𝑥)

In the case of 𝑔(𝑥) =0, the ODE is separable. Sometimes this is written as

𝑃(𝑥)𝑑𝑦𝑑𝑥+𝑄(𝑥)𝑦=𝐺(𝑥)

which can be rearranged trivially to yield the standard form.

Solutions

• Integrable LHS
• Integrating factor
  • [[#Integrating factor#Explanation|Explanation]]
  • [[#Integrating factor#Algorithm|Algorithm]]

Integrable LHS

In some cases a first-order linear ODE may be immediately solvable by integration. This is the case when the LHS is or can be rearranged to be the case of a product rule, that is of the form

𝑃(𝑥)𝑑𝑦𝑑𝑥+𝑃′(𝑥)𝑦=𝐺(𝑥)

which can easily be rewritten as

𝑑𝑑𝑥(𝑃(𝑥)𝑦)=𝐺(𝑥)⟹𝑃(𝑥)𝑦=∫𝐺(𝑥)𝑑𝑥

so the general solution is

𝑦=∫𝐺(𝑥)𝑑𝑥𝑃(𝑥)

including the integration constant. In cases where this is not possible, an Integrating factor 𝜇(𝑥) may be used to make it so.

Integrating factor

Explanation

In cases where the LHS is not immediately integrable, it may be made so by multiplying both sides by a chosen integrating factor, conventionally called either 𝜇(𝑥) or 𝐼(𝑥)².

𝜇(𝑥)𝑑𝑦𝑑𝑥+𝜇(𝑥)𝑝(𝑥)𝑦=𝜇(𝑥)𝑔(𝑥)

We select 𝜇(𝑥) such that the LHS is the result of a product rule, i.e.

𝑑𝑑𝑥(𝜇(𝑥)𝑦)=𝜇(𝑥)𝑑𝑦𝑑𝑥+𝜇(𝑥)𝑝(𝑥)𝑦⟹𝜇′(𝑥)=𝑝(𝑥)𝜇(𝑥)

A family of solutions to this very basic differential equation for 𝜇 is given by

𝜇(𝑥)=exp⁡(∫𝑝(𝑥)𝑑𝑥)

where the integration constant may be conveniently selected as 0. Thus, the ODE may be solved generally

𝑑𝑑𝑥(𝜇(𝑥)𝑦(𝑥))=𝜇(𝑥)𝑔(𝑥)⟹𝑦(𝑥)=1𝜇(𝑥)∫𝜇(𝑥)𝑔(𝑥)𝑑𝑥

Algorithm

Algorithmically, the integrating factor method is summarised as follows

1. Write the differential equation in standard form
2. Find the integrating factor (setting the constant of integration conveniently to 0)

𝜇(𝑥)=exp⁡(∫𝑝(𝑥)𝑑𝑥)

3. The general solution is

𝑦(𝑥)=1𝜇(𝑥)∫𝜇(𝑥)𝑔(𝑥)𝑑𝑥+𝐶𝜇(𝑥)

Practice problems

• 2017. Elementary differential equations and boundary value problems, pp. 31–32 (§2.1 problems)

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Footnotes

1. 2017. Elementary differential equations and boundary value problems, pp. 24ff. (§2.1) ↩

2. I will be using the former since it is prettier. ↩