In this section we will treat equations of the following two kinds:
and changes of variables that are sometimes used to solve some particular cases of first-degree differential equations. This section is not absolutely formal. Its purpose is to try to get methods to solve certain types of equations and study variable changes that may become interesting. Before reading this section, It would be better to start reading Integrating Factors in Differential Equations of First Order.
In this section, we are going to try to solve Bernoulli differential equations and Riccati differential equations, but firstly let's deal with some variable changes and underlying theory that is involved, to address these problems.
A. Let's first consider a change of form , i.e. In particular, equation becomes If is the family of solutions of this equation for and , then is the family of solutions of for and . Later, clearing in and substituting it in , we'll get a formula for the family of solutions of .
We'll use this change in Riccati differential equation later.
B. Other times, we simply can do and substituting in we get If is the solution of this equation for and , then the solution of equation for and is .
C. Making the change and with the change we get Then becomes .
Conclusion: If is the solution of this equation for and , then the solution for equation for and is
We'll use this change for Bernouilli differential equation later.
Riccati Differential Equation:
One differential equation of first order is named a Riccati differential equation if it is of the form
Note: If , then it becomes a differential linear equation of first order, and it can be (in general) dealt with integrating factors. If , it will become a Bernoulli differential equation, which we are going to deal with later. Let's suppose we know a particular solution of . Then the variable change
transforms into a Bernoulli differential equation with , and then the change transforms into a differential equation of first order.
Solve the Riccati equation .
One particular solution of this equation is , and making the change , we'll get and the original equation is transformed into
which is a linear differential equation of first order with solution
Therefore, the general solution of this equation is
Bernoulli Differential Equation:
One differential equation of first order is named a Bernouilli differential equation if it is of the form
When or , is a linear differential equation. When and , we can make the change
and this lets us transform equation into a linear differential equation of first order. Indeed,
and multiplying by gives
For a more complete explanation for this change of variable with an example and exercise, visit Bernouilli equation.
D. The second order differential equation can be solved making the change of variable and, later, if we get a solution for , it will be sufficient to integrate to solve the initial equation.
Prove this change with the following exercise:
Relevant wiki: Separable Differentiable Equations
If is a solution of then is a solution of
Knowing that , because , and hence . Therefore,
Authors: Silvia Novo, Rafael Obaya,Jesús Rojo. Authors: Edwards, Penney.