Question

Find nine solutions of the initial value problem $$ y^{\prime}=3 x(y-1)^{1 / 3}, \quad y(0)=1 $$ that are all defined on \((-\infty, \infty)\) and differ from each other for values of \(x\) in every open interval that contains \(x_{0}=0 .\)

Short answer

Question: Find nine different solutions for the given initial value problem (IVP) that are defined for all real values of x and differ from each other for values of x in every open interval containing x_0 = 0. IVP: $$ \frac{dy}{dx}=3 x(y-1)^{1 / 3}, \quad y(0) = 1 $$ Answer: The nine different solutions for the given IVP that satisfy the conditions are given by: $$ y_i(x)=1 + \left(\frac{x^2 - 8 / 27}{3}\right)^{3/2} \quad i = 1, 2, \dots, 9 $$

Step-by-step solution

Analyze the given differential equation

The given differential equation can be written as: $$ \frac{dy}{dx}=3 x(y-1)^{1 / 3} $$ This is a separable first-order differential equation, where we can separate variables y and x on both sides of the equation.

Separate variables

To separate the variables, we divide both sides by \((y-1)^{1 / 3}\) and integrate with respect to y and x: $$ \frac{dy}{(y-1)^{1 / 3}}=3x dx $$

Integrate both sides

Now, we integrate both sides with respect to their respective variables: $$ \int \frac{dy}{(y-1)^{1 / 3}}=\int 3x dx $$

Evaluate the integrals

After evaluating the integrals, we have: $$ 3(y-1)^{2 / 3} = x^2 + C $$ where C is the constant of integration.

Apply the initial condition y(0)=1

Substitute the initial condition into the equation and solve for C: $$ 3(1-1)^{2 / 3} = 0^2 + C \Rightarrow C = 0 $$

Write the particular solution

Using the value of C we found, the particular solution of the given IVP is: $$ 3(y-1)^{2/3} = x^2 $$

Find nine solutions

According to the problem, we are asked to find nine different solutions that satisfy the conditions. As the given IVP is an autonomous equation with the relation between x and y as y'=f(y), we can use the level curves following different values for the constant C: $$ 3(y-1)^{2/3} = x^2 + C_i $$ When \(C_i = -\frac{8}{27}\), the equation becomes: $$ x^2 - \frac{8}{27} = 3(y-1)^{2/3} $$ which has nine distinct solution curves (3 positive \(y\) value curves, 3 negative \(y\) value curves, and 3 corresponding to horizontal lines for negative \(x\) values): $$ y_i(x)=1 + \left(\frac{x^2 - 8 / 27}{3}\right)^{3/2} \quad i = 1, 2, \dots, 9 $$

Key concepts

First-Order Differential Equations

First-order differential equations are a fundamental class of ordinary differential equations (ODEs) where the unknown function's highest derivative is of the first order. In mathematical terms, a first-order ODE has the general form \( \frac{dy}{dx} = f(x, y) \), where \( f(x, y) \) is a given function of two variables. To solve these equations, we aim to find an unknown function \( y(x) \) that satisfies the given differential equation for all \( x \) in a certain interval.

For the initial value problem (IVP), we are given an additional piece of information: the value of the unknown function at a specific point, commonly denoted as \( y(x_0) = y_0 \). The goal here is not only to find a function that satisfies the differential equation but also to match the given initial condition. The presence of an initial condition allows us to find a unique solution, which is especially important when the general solution contains arbitrary constants.

Understanding the Basics

In our example, we have the first-order ODE \( y' = 3x(y - 1)^{1/3} \), with the initial condition \( y(0) = 1 \). To obtain solutions, we would first need to consider the type of the first-order equation—whether it is separable, linear, or autonomous, for example—and then apply appropriate techniques to reach the solution that satisfies both the differential equation and the initial condition.

Separable Differential Equations

Separable differential equations form a subcategory of first-order ODEs, characterized by the ability to algebraically manipulate the equation so that all the terms involving the independent variable \( x \) are on one side, and all the terms involving the dependent variable \( y \) are on the other side. The general form of a separable equation is \( \frac{dy}{dx} = g(x)h(y) \), which can be rearranged to \( \frac{1}{h(y)}dy = g(x)dx \).

The strategy for solving separable equations involves writing the differential equation in this separated form and then integrating both sides to find \( y \) as a function of \( x \), often introducing an arbitrary constant \( C \) along the way.

Step-by-Step Process

In our exercise, we manipulated the given ODE into a separable form \( \frac{dy}{(y-1)^{1/3}} = 3x dx \). Following different values of the integration constant \( C \), this approach yields multiple solutions, exemplifying the variety of functions that can arise from a single differential equation.

Separable equations are particularly convenient because the method of separation and integration is straightforward and mechanical, allowing for a clear pathway from the problem to the solution.

Autonomous Differential Equations

Autonomous differential equations are a specific type of first-order ODE where the rate of change of the unknown function does not directly depend on the independent variable—the time or position, often denoted as \( x \) in general ODE contexts. Mathematically, an autonomous ODE is of the form \( y' = f(y) \), highlighting the absence of the independent variable in the function \( f \).

What's interesting about autonomous equations is that they often represent time-invariant processes where the state of the system at any given point depends solely on its current state, not on when that state is achieved.

Exploring Multiple Solutions

The problem presented involves finding multiple solutions to an autonomous differential equation. By treating the constant of integration, \( C_i \), as a parameter that can take various values, we establish a family of function curves that can qualify as potential solutions to the IVP. This is demonstrated in the final step of our example where different constants modify the equation, leading to unique solution curves for each chosen value.

This highlights the influence of the initial conditions and constants of integration in determining a particular solution to an autonomous differential equation. By considering different scenarios through a range of constants, multiple valid solutions—meeting the requirements of the IVP—can be obtained.