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Chapter 3: Problem 2

Solve the differential equation $$ \frac{d w}{d z}=w-w^{2} $$ Show that it has poles as its only singularity.

Short Answer

Expert verified
The differential equation's only singularities are poles.

Step by step solution

01

Identify the Differential Equation

The given differential equation is \ \( \frac{d w}{d z} = w - w^2 \). The goal is to analyze the nature of its singularities.
02

Separate Variables

Rewrite the equation to separate the variables w and z. Rearrange it to: \ \( \frac{1}{w - w^2} \frac{d w}{d z} = 1 \).
03

Simplify the Integration

Express \( \frac{1}{w - w^2} \) in a more integrable form by partial fraction decomposition: \ \( \frac{1}{w(1 - w)} = \frac{A}{w} + \frac{B}{1 - w} \). Solving for A and B yields \( A = 1 \) and \( B = 1 \). \ Hence, \( \frac{1}{w(1 - w)} = \frac{1}{w} + \frac{1}{1 - w} \).
04

Integrate Both Sides

Integrate both sides with respect to z: \ \( \int \left( \frac{1}{w} + \frac{1}{1 - w} \right) dw = \int 1 dz \). After integrating, we get: \ \( \ln|w| - \ln|1 - w| = z + C \), where C is the constant of integration.
05

Combine the Logarithmic Terms

Combine the logarithms: \ \( \ln \left| \frac{w}{1 - w} \right| = z + C \). Exponentiate both sides to solve for w: \ \( \left| \frac{w}{1 - w} \right| = e^{z + C} \).
06

Solve for W

Let \( A = e^C \), then \( \frac{w}{1 - w} = \pm Ae^z \). Solve for w: \ \( w = \frac{A e^z}{1 + A e^z} \) or \( w = \frac{-A e^z}{1 - A e^z} = \frac{A e^z}{A e^z - 1} \).
07

Determine Singularity Types

Examine the solutions for singularities. For \( w = \frac{A e^z}{1 + A e^z} \), singularities occur when the denominator is zero, i.e., \( 1 + A e^z = 0 \). Hence, \( e^z = -\frac{1}{A} \), which is not possible since the exponential function is never negative. For \( w = \frac{-A e^z}{1 - A e^z} \), singularities occur when \( 1 - A e^z = 0 \), i.e., \( e^z = \frac{1}{A} \). This is possible, hence we have a pole when \( z = \ln \frac{1}{A} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Singularities
In the context of differential equations, singularities are points where the solution to the differential equation becomes undefined or unbounded. These points are critical in understanding the behavior of the system described by the differential equation. For the given differential equation: \[ \frac{d w}{d z} = w - w^2 \], we aim to find such points. Singularities are often identified as poles, zeroes, or essential singularities. In this problem, we are specifically looking for poles, which are points where the function approaches infinity. By analyzing the solutions, we determine the singularities by finding when the denominator of the expression equals zero, resulting in an undefined function.
Partial Fraction Decomposition
Partial fraction decomposition is a method used to break down complex rational functions into simpler fractions, making integration easier. In our problem, we start with the expression: \[ \frac{1}{w - w^2} \], which can be decomposed as: \[ \frac{1}{w(1 - w)} \]. By expressing this in terms of partial fractions, we get: \[ \frac{1}{w(1 - w)} = \frac{A}{w} + \frac{B}{1 - w} \]. Solving for A and B, both are found to be 1, so the expression simplifies to: \[ \frac{1}{w} + \frac{1}{1 - w} \]. This simplified form makes it easier to integrate the expression and solve the differential equation step-by-step.
Separation of Variables
Separation of variables is a fundamental technique for solving differential equations. It involves rewriting the equation to isolate the variables on opposite sides of the equation. This allows us to integrate each side with respect to its own variable. For the given differential equation: \[ \frac{d w}{d z} = w - w^2 \], we separate variables by rearranging it as: \[ \frac{1}{w - w^2} \frac{d w}{d z} = 1 \]. After decomposing \(\frac{1}{w - w^2} \) using partial fractions and simplifying, we can integrate both sides to find the general solution.
Pole Analysis
Analyzing poles involves finding when the function becomes unbounded. In our solution, once we reach: \[ w = \frac{A e^z}{1 + A e^z} \] and \[ w = \frac{-A e^z}{1 - A e^z} = \frac{A e^z}{A e^z - 1} \], we identify poles by checking when the denominators are zero. For the first part \(\frac{A e^z}{1 + A e^z} \), the denominator is never zero because \( e^z \) is never negative. Thus, no poles exist here. However, for \( \frac{-A e^z}{1 - A e^z} \), singularities occur when \( e^z = \frac{1}{A} \). This leads to poles at \( z = \text{ln} (\frac{1}{A}) \), highlighting points where the solution becomes infinite.

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Most popular questions from this chapter

Discuss the nature of the singular points (location, fixed, or movable) of the following differential equations and solve the differential equations. (a) \(z \frac{d w}{d z}=2 w+z\) (b) \(z \frac{d w}{d z}=w^{2}\) (c) \(\frac{d w}{d z}=a(z) w^{3}, \quad a(z)\) is an entire function of \(z\). (d) \(z^{2} \frac{d^{2} w}{d z^{2}}+z \frac{d w}{d z}+w=0\)

Let \(\Gamma(z)\) be given by $$ \frac{1}{\Gamma(z)}=z e^{\gamma z} \prod_{n=1}^{\infty}\left(1+\frac{z}{n}\right) e^{-z / n} $$ for \(z \neq 0,-1,-2, \ldots\) and \(\gamma=\) constant. (a) Show that $$ \frac{\Gamma^{\prime}(z)}{\Gamma(z)}=-\frac{1}{z}-\gamma-\sum_{n=1}^{\infty}\left(\frac{1}{z+n}-\frac{1}{n}\right) $$ (b) Show that $$ \frac{\Gamma^{\prime}(z+1)}{\Gamma(z+1)}-\frac{\Gamma^{\prime}(z)}{\Gamma(z)}-\frac{1}{z}=0 $$ whereupon $$ \Gamma(z+1)=C z \Gamma(z), \quad C \text { constant } $$ (c) Show that \(\lim _{z \rightarrow 0} z \Gamma(z)=1\) to find that \(C=\Gamma(1)\). (d) Determine the following representation for the constant \(\gamma\) so that \(\Gamma(1)=1\) $$ e^{-\gamma}=\prod_{n=1}^{\infty}\left(1+\frac{1}{n}\right) e^{-1 / n} $$ (e) Show that $$ \prod_{n=1}^{\infty}\left(1+\frac{1}{n}\right) e^{-1 / n}=\lim _{n \rightarrow \infty} \frac{2}{1} \frac{3}{2} \frac{n+1}{n} e^{-S(n)}=\lim _{n \rightarrow \infty}(n+1) e^{-S(n)} $$ where \(S(n)=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}\). Consequently obtain the limit $$ \gamma=\lim _{n \rightarrow \infty}\left(\sum_{k=1}^{n} \frac{1}{k}-\log (n+1)\right) $$ The constant \(\gamma=0.5772157 \ldots\) is referred to as Euler's constant.

Suppose we are given the equation \(d^{2} w / d z^{2}=2 w^{3}\). (a) Let us look for a solution of the form $$ w=\sum_{n=0}^{\infty} a_{n}\left(z-z_{0}\right)^{n-r}=a_{0}\left(z-z_{0}\right)^{-r}+a_{1}\left(z-z_{0}\right)^{1-r}+\cdots $$ for \(z\) near \(z_{0}\). Substitute this into the equation to determine that \(r=1\) and \(a_{0}=\pm 1\) (b) "Linearize" about the basic solution by letting \(w=\pm 1 /\left(z-z_{0}\right)+v\) and dropping quadratic terms in \(v\) to find \(d^{2} v / d z^{2}=6 v /\left(z-z_{0}\right)^{2}\). Solve this equation (Cauchy-Euler type) to find $$ v=A\left(z-z_{0}\right)^{-2}+B\left(z-z_{0}\right)^{3} $$ (c) Explain why this indicates that all coefficients of subsequent powers in the following expansion (save possibly \(a_{4}\) ) $$ w=\frac{\pm 1}{\left(z-z_{0}\right)}+a_{1}+a_{2}\left(z-z_{0}\right)+a_{3}\left(z-z_{0}\right)^{2}+a_{4}\left(z-z_{0}\right)^{3}+\cdots $$ can be solved uniquely. Substitute the expansion into the equation for \(w\), and find \(a_{1}, a_{2}\), and \(a_{3}\), and establish the fact that \(a_{4}\) is arbitrary. We obtain two arbitrary constants in this expansion: \(z_{0}\) and \(a_{4}\). The solution to \(w^{\prime \prime}=2 w^{3}\) can be expressed in terms of elliptic functions; its general solution is known to have only simple poles as its movable singular points. (d) Show that a similar expansion works when we consider the equation $$ \frac{d^{2} w}{d z^{2}}=z w^{3}+2 w $$ (this is the second Painlev茅 equation (Ince, 1956)), and hence that the formal analysis indicates that the only movable algebraic singular points are poles. (Painlev茅 proved that there are no other singular points for this equation.) (e) Show that this expansion fails when we consider $$ \frac{d^{2} w}{d z^{2}}=2 w^{3}+z^{2} w $$ because \(a_{4}\) cannot be found. This indicates that a more general expansion is required. (In fact, another term of the form \(b_{4}\left(z-z_{0}\right)^{3} \log \left(z-z_{0}\right)\) must be added at this order, and further logarithmic terms must be added at all subsequent orders in order to obtain a consistent formal expansion.)

Demonstrate whether or not the following sequences are Cauchy sequences: (a) \(\left\\{z^{n}\right\\}_{n=1}^{\infty}, \quad|z|<1\) (b) \(\left\\{1+\frac{z}{n}\right\\}_{n=1}^{\infty}, \quad|z|<\infty\) (c) \(\\{\cos n z\\}_{n=1}^{\infty}, \quad|z|<\infty\) (d) \(\left\\{e^{-n / z}\right\\}_{n=1}^{\infty}, \quad|z|<1\)

Show that if \(f(z)\) is meromorphic in the finite \(z\) plane, then \(f(z)\) must be the ratio of two entire functions.
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