Question

Substitute \(v=y^{\prime} / y\) in the homogeneous SOLDE $$ y^{\prime \prime}+p(x) y^{\prime}+q(x) y=0 $$ and: (a) Show that it turns into \(v^{\prime}+v^{2}+p(x) v+q(x)=0\), which is a firstorder nonlinear equation called the Riccati equation. Would the same Riccati equation substitution work if the DE were inhomogeneous? (b) Show that by an appropriate transformation, the Riccati equation can be directly cast in the form \(u^{\prime}+u^{2}+S(x)=0\).

Short answer

a) Substituting \(v = y' / y\) into the homogeneous ODE transforms the equation into a Riccati equation: \(v' + v^2 + p(x)v + q(x) = 0\). The same transformation does not work for an inhomogeneous DE. b) By using transformation \(v = -u' / u\), the Riccati equation can be directly cast in the form \(u' + u^2 + S(x) = 0\).

Step-by-step solution

Understanding the given problem

Firstly, understand the given Ordinary Differential Equation (ODE). The ODE given is a second order linear ODE. The task is to substitution \(v = y' / y\) into the given ODE. The substituted equation should transform the given equation into a first-order non-linear equation also known as a Riccati Equation which is in the form \(v' + v^2 + p(x)v + q(x) = 0\).

Substitution into homogeneous DE

The next step is to substitute \(v = y' / y\) into the homogeneous ODE. Substitute \(y'\) as \(vy\) and \(y''\) as \(v' \cdot y + v^2 \cdot y\). This will lead to the equation becoming \(v' \cdot y + v^2 \cdot y + p(x) \cdot vy + q(x) \cdot y = 0\). Since \(y \neq 0\), one can remove \(y\) from the equation, converting it into a Riccati equation \(v' + v^2 + p(x) \cdot v + q(x) = 0\).

Check if substitution works for an inhomogeneous DE

To investigate if the same transformation works for an inhomogeneous DE, suppose we have an inhomogeneous DE \(y'' + p(x)y' + q(x)y = f(x)\) where \(f(x)\) is some known function. Substitute \(y'\) as \(vy\) and \(y''\) as \(v'y + v^2y\). This changes the DE into \(v'y + v^2y + p(x)vy + q(x)y = f(x)\). It can be seen that due to the \(f(x)\) term, the same substitution will not yield a Riccati equation.

Transformation of the Riccati equation

Let's consider a transformation \(v = -u' / u\). Substituting it into the Riccati equation gives \(-(u'' / u) + u'^2 / u^2 - p(x)u' / u - q(x) = 0\). Multiply by \(-u^2\), we get the equation as \(u'' + u^2 + p(x)u' + q(x)u^2 = 0\). Here we can denote \(S(x) = p(x) + u(x)\). Which yields the equation as \(u' + u^2 + S(x) = 0\).

Key concepts

Ordinary Differential Equation

An Ordinary Differential Equation (ODE) is a mathematical expression involving a function and its derivatives. In essence, it's an equation that relates some unknown function, typically denoted as 'y' or 'f', to its derivatives, representing the rates at which these functions change. These equations can range from simple, first-order one-variable equations to complex, higher-order systems.

When tackling an ODE, it's essential to identify its order, which corresponds to the highest derivative present in the equation, and whether it's linear or nonlinear. Linear ODEs have solutions that form a straight line when graphed and do not have terms like the square or product of the function or its derivatives. Nonlinear ODEs, including the Riccati equation, have more complex behaviors and do not simply form straight lines when graphed.

Substitution Method

The substitution method is a strategy often employed to simplify differential equations, making them easier to solve. It involves replacing an expression with a new variable to reduce the complexity of the original equation. This is akin to finding a simpler path through a complicated maze - by cleverly choosing what to substitute, one can turn a difficult problem into a more manageable one.

For example, substituting \(v = y' / y\) simplifies a second-order ODE by expressing it in terms of the new variable, \(v\). Essentially, we are transforming the viewpoint from which we approach the problem, often revealing a different aspect of the equation's structure which can then be exploited to find a solution.

Inhomogeneous Differential Equation

An inhomogeneous differential equation is one that includes a term independent of the function and its derivatives. This non-homogeneous term, which we can denote as \(f(x)\), adds an extra layer of complexity to the equation since it represents an external input or influence on the system being modeled.

In terms of finding solutions, while homogeneous equations allow for the superposition of solutions, inhomogeneous equations typically require particular solutions that account for the non-homogeneous term. This means that the strategies applicable to homogeneous equations, like the substitution method used for the Riccati equation, may not directly transfer to inhomogeneous cases without modifications.

Linear ODE

A linear Ordinary Differential Equation (Linear ODE) depicts a relationship in which the unknown function and its derivatives appear to the first power and are not multiplied by each other—no terms like \(y^2\) or \(yy'\), for instance. They are akin to straight lines, and their general form for an nth-order linear ODE is \[ a_{n}(x)y^{(n)} + a_{n-1}(x)y^{(n-1)} + ... + a_{1}(x)y' + a_{0}(x)y = g(x) \], where \(a_{n}(x)\) are coefficient functions and \(g(x)\) is the inhomogeneous part.

Linear ODEs are well-understood, characterized by well-established methods for finding their solutions. Their superposition principle, which states that the sum of two solutions is also a solution, simplifies their analysis and the determination of particular and general solutions.

Nonlinear ODE

Nonlinear Ordinary Differential Equations (Nonlinear ODEs) differ from their linear counterparts in that they can contain powers or products of the unknown function and its derivatives. The complexities introduced by these terms can lead to diverse behaviors in their solutions, such as chaotic behavior or multiple equilibrium points.

The Riccati equation is a classic example of a first-order Nonlinear ODE, with the general form being \[ v' + v^2 + p(x)v + q(x) = 0 \]. Despite being more challenging to solve, certain substitution methods can transform these nonlinear ODEs into more tractable forms or even linear ODEs, facilitating the finding of solutions. Nevertheless, each nonlinear ODE is unique, and a method that works for one may not necessarily be applicable to another.