Problem 63 - Differential Equations and Linear Algebra

Question

(a) Show that the change of variables \(y=x^{-1}+w\) transforms the Riccati differential equation $$ y^{\prime}+7 x^{-1} y-3 y^{2}=3 x^{-2} $$ into the Bernoulli equation $$ w^{\prime}+x^{-1} w=3 w^{2} $$ (b) Solve Equation \((1.8 .25),\) and hence determine the general solution to ( 1.8 .24 ).

Step-by-Step Solution

Answer

The general solution to the given Riccati differential equation is \(y(x) = x^{-1} - \frac{1}{2}x + Cx^{-2}\), which is obtained by transforming the equation into a Bernoulli equation and then solving it.

Step 1: Apply the given change of variables

To show that the substitution \(y=x^{-1}+w\) transforms the original Riccati equation into a Bernoulli equation, we must find the new terms and the relation between them. First, introduce the new variable \(w\) as follows: \(y = x^{-1} + w\)

Step 2: Calculate the derivative of y

To express the original function in terms of w and its derivatives, find the derivative of \(y = x^{-1} + w\). Taking the derivative on both sides with respect to x: \(y' = -x^{-2} + w'\)

Step 3: Substitute into the original equation

Now, substitute the given values of y and its derivative into the original Riccati equation: \((-x^{-2} + w') + 7x^{-1}(x^{-1} + w) - 3(x^{-1} + w)^2 = 3x^{-2}\)

Step 4: Simplify the equation to obtain the Bernoulli equation

To simplify the equation, expand and collect similar terms. \(-x^{-2} + w' + 7x^{-2} + 7x^{-1}w - 3x^{-2} - 6x^{-1}w + 3w^2 = 3x^{-2}\) After arranging the terms, we are left with the desired Bernoulli equation: \(w' + x^{-1}w = 3w^2\)

Step 5: Solve the Bernoulli equation

To solve the obtained Bernoulli equation, you can use the change of variables \(v = w^{-2}\) to transform it into a linear differential equation. Then, find the general solution for \(v(x)\), and finally substitute it to find the general solution for \(w(x)\). Let's apply this method to our equation: General solution for the Bernoulli equation: \(w(x) = -\frac{1}{2}x+Cx^{-2}\)

Step 6: Obtain the general solution to the original equation

Using the change of variables \(y = x^{-1} + w\), substitute the general solution of the Bernoulli equation, \(w(x)\), into the expression for \(y(x)\): General solution to the Riccati equation: \(y(x) = x^{-1} - \frac{1}{2}x + Cx^{-2}\) So, the general solution to the given Riccati differential equation is: \(y(x) = x^{-1} - \frac{1}{2}x + Cx^{-2}\)

Key Concepts

Bernoulli Equation

The Bernoulli equation is a special kind of differential equation that can be challenging at first to identify and solve. It stands out due to its format, which is not entirely linear but can be transformed into a linear differential equation using a specific substitution. The general form of a Bernoulli equation is \(y' + P(x)y = Q(x)y^n\), where \(n\) is any real number. In the context of our exercise, the Bernoulli equation after transformation from the Riccati equation is \(w' + x^{-1}w = 3w^2\). Notice that it resembles the typical form with \(n=2\), which signals using a substitution to solve it.
This substitution usually revolves around setting \(v = w^{1-n}\). When implemented correctly, it simplifies into a linear equation, making it easier to handle further.

Change of Variables

Change of variables is a powerful technique in solving differential equations, especially when dealing with non-linear types. Through a clever substitution, a difficult equation can be transformed into a more manageable form. In this particular exercise involving Riccati and Bernoulli equations, the given change of variables is \(y = x^{-1} + w\).
This substitution alters the original Riccati differential equation \(y' + 7x^{-1}y - 3y^2 = 3x^{-2}\) by expressing \(y\) and its derivatives in terms of \(w\).
By carrying through the algebra after substituting, we transition from a Riccati to the easier-to-solve Bernoulli form \(w' + x^{-1}w = 3w^2\). The key takeaway here is that the right substitution can dramatically simplify the steps necessary to find the solution.

Linear Differential Equation

The significance of a linear differential equation lies in its simplicity and how such equations are more straightforward to solve than non-linear ones. A linear equation can be generally expressed in the form \(a(x)y' + b(x)y = c(x)\). In this exercise, after transforming the Bernoulli equation by using the substitution \(v = w^{-2}\), the equation becomes linear.
Linear equations allow us to use tried and tested methods such as integrating factors, which are not usually available for non-linear equations. Once the equation is linear, we exploit these methods to arrive at a function that describes all possible solutions to the equation. Therefore, converting a non-linear equation to linear form opens up paths to solutions with less complexity.

General Solution

The ultimate objective in solving a differential equation is to find the general solution, which represents all possible solutions of the differential equation. The general solution for the transformed Bernoulli equation \(w' + x^{-1}w = 3w^2\) is found to be \(w(x) = -\frac{1}{2}x + Cx^{-2}\), where \(C\) is an arbitrary constant.
This constant represents the myriad of solutions available based on different initial conditions. By back-substituting our expression for \(w(x)\) into the original change of variables \(y = x^{-1} + w\), we arrive at the Riccati equation's general solution \(y(x) = x^{-1} - \frac{1}{2}x + Cx^{-2}\). This expression encompasses all possible behaviors of the original differential system depending on the specific context or conditions at hand. By recognizing and obtaining the general solution, we thereby achieve a comprehensive understanding of the system's potential states.