Question

The nonlinear first order equation $$y^{\prime}+r(x) y^{2}+p(x) y+q(x)=0$$ is the generalized Riccati equation. (See Exercise 2.4.55.) Assume that \(p\) and \(q\) are continuous and \(r\) is differentiable. (a) Show that \(y\) is a solution of (A) if and only if \(y=z^{\prime} / r z,\) where $$z^{\prime \prime}+\left[p(x)-\frac{r^{\prime}(x)}{r(x)}\right] z^{\prime}+r(x) q(x) z=0$$ (b) Show that the general solution of \((\mathrm{A})\) is $$y=\frac{c_{1} z_{1}^{\prime}+c_{2} z_{2}^{\prime}}{r\left(c_{1} z_{1}+c_{2} z_{2}\right)}$$ where \(\left\\{z_{1}, z_{2}\right\\}\) is a fundamental set of solutions of (B) and \(c_{1}\) and \(c_{2}\) are arbitrary constants.

Short answer

Question: Prove that a function y is a solution of the generalized Riccati equation if and only if it is equal to (z's derivative / rz), with z satisfying a second-order linear differential equation. Also, show that the general solution of the Riccati equation can be expressed in terms of two arbitrary constants and a fundamental set of solutions of the second-order linear differential equation. Answer: We derived the second-order linear differential equation for z using the relation between y and z (given in the problem). Also, we proved that the general solution of the Riccati equation can be represented using two arbitrary constants and the fundamental set of solutions for the derived second-order linear differential equation for z.

Step-by-step solution

Derive the equation for z

Let's start by considering the relation between y and z given in (a): $$y=\frac{z^{\prime}}{rz}$$. Let's find the derivative of y (denoted as y') using this relation. Using the quotient rule, we get: $$y^{\prime}=\frac{z^{\prime \prime}*rz-z^{\prime}*r^{\prime}*z-z^{\prime 2}}{(rz)^{2}}$$. Now, let's plug this expression for y' into the Riccati equation: $$\frac{z^{\prime \prime}*rz-z^{\prime}*r^{\prime}*z-z^{\prime 2}}{(rz)^{2}} + ry^2 + py + q = 0$$ Substitute $$y=\frac{z^{\prime}}{rz}$$ in the above equation and simplify: $$\frac{z^{\prime \prime} * rz - z^{\prime} * r^{\prime} * z - z^{\prime 2} + r^2 * z^{\prime 2} + p * r * z^{\prime} * z + r * q * z^2}{(rz)^2} = 0$$. This equation is satisfied if the numerator is zero: $$z^{\prime\prime}*rz - z^{\prime}*r^{\prime}*z - z^{\prime 2} + r^2 *z^{\prime 2} + p * r * z^{\prime} * z + r * q * z^2=0$$ Rearranging the terms, we recover the equation for z given in (a): $$z^{\prime\prime}+\left[p(x)-\frac{r^{\prime}(x)}{r(x)}\right]z^{\prime}+r(x)q(x)z=0$$

Prove the general form of the solution for the Riccati equation using the fundamental set of the linear equation for z

We know that the general solution for the second-order linear equation for z has the form $$z = c_1*z_1 + c_2*z_2$$, where $$z_1$$ and $$z_2$$ are the fundamental set of solutions and $$c_1, c_2$$ are arbitrary constants. To show that the general solution of the Riccati equation, we need to plug this value for z into the relation between z and y: $$y = \frac{z^{\prime}}{rz}$$. First, let's find the derivative of z: $$z^{\prime} = c_1 * z_1^{\prime} + c_2 * z_2^{\prime}$$. Now plug this into the relation: $$y = \frac{c_1 * z_1^{\prime} + c_2 * z_2^{\prime}}{r(c_1 * z_1 + c_2 * z_2)}$$. Thus, we've demonstrated that the general solution of the Riccati equation is of the form given in (b).

Key concepts

Nonlinear Differential Equations

Differential equations are mathematical equations that relate some function with its derivatives. These equations can describe various real-life situations, such as physical systems. However, these equations come in different forms, and one of the types is nonlinear differential equations. Nonlinear equations are distinguished by the fact that the unknown function or its derivatives appear with exponents other than one. This nonlinear behavior often makes these equations more tricky to solve compared to their linear counterparts.

In the context of Riccati equations, we find them as a subcategory of first-order, nonlinear differential equations. Their form is typically expressed as:

• They contain products of the unknown function and its derivative, leading to a non-linear relationship between terms.
• The generalized Riccati equation introduced in the exercise shows a complexity that arises due to involvement of terms like \(ry^2\).
• The nature of these equations often requires specific techniques or transformations for solutions, like the substitution method used with the function \(z\).

Remember, the presence of nonlinear terms makes the general methodology for solving these equations challenging yet fascinating.

First Order Differential Equations

First-order differential equations describe a wide range of dynamics, often capturing the rate of change of a system at a single point in time. The order of a differential equation refers to the highest derivative that the equation contains, so first-order means the highest is the first derivative. These equations are fundamental in understanding rates and initial behaviors in dynamic systems.

The primary structure of a first-order differential equation is \(y' = f(x, y)\), where \(y'\) is the first derivative of \(y\) with respect to \(x\). In the case of the Riccati equation, it is fascinating because its solution still needs a deeper insight into its non-linear nature:

• The term \(y\) and its derivatives appear in the same function, contributing to the difficulty.
• By transforming these equations using specific techniques, one can often address the nonlinearity for further analysis, as is done with \(y = \frac{z'}{rz}\).

These equations are pivotal as they allow us to model and solve problems that feature continuous change, further seeking an impactful link to physical phenomena.

General Solution

A general solution in the context of differential equations refers to a solution that encompasses all possible particular solutions. This is typically expressed in terms of arbitrary constants that adjust to meet initial or boundary conditions specific to the case at hand. For differential equations like the Riccati, finding a general solution requires unveiling all behavior permutations displayed by its solutions.

In the exercise, we see that the general solution is expressed in the form \(y = \frac{c_1 z_1' + c_2 z_2'}{r(c_1 z_1 + c_2 z_2)}\), where:

• \(c_1\) and \(c_2\) are arbitrary constants. These constants enable solving the equation under various conditions.
• The terms \(z_1\) and \(z_2\) signify the fundamental set of solutions essential for constructing any specific solution.

This general solution is crucial for exploring the entire behavior of the system setup by the differential equations and identifying particular solutions fitting specific contexts.

Fundamental Set of Solutions

The concept of a fundamental set of solutions is essential when tackling linear differential equations, especially when constructing a complete, general solution. It typically involves finding a set of linearly independent solutions from which every possible solution can be constructed.

In simpler terms, the fundamental set forms the basis for solutions to the differential equation. In the provided exercise, for the corresponding second-order linear equation in \(z\), \(z_1\) and \(z_2\) constitute such a set:

• These solutions are independent, meaning one cannot be represented as a linear combination of another.
• The general solution for the equation can be expressed in terms of these fundamental solutions, incorporating arbitrary constants.

Understanding and working with these sets is pivotal in navigating through complex systems of differential equations as they anchor the path toward solving general solutions and, consequently, any specific solution that adheres to the system's initial or boundary conditions.