Problem 10 - Calculus: Early Transcendentals

Question

Find a non-constant solution of the initial value problem \(y^{\prime}=y^{1 / 3}, y(0)=0,\) using separation of variables. Note that the constant function \(y(t)=0\) also solves the initial value problem. This shows that an initial value problem can have more than one solution.

Step-by-Step Solution

Answer

The non-constant solution is \( y = \left(\frac{4}{3}t\right)^{3/4} \).

Step 1: Set Up the Differential Equation

We start with the given differential equation: \( y' = y^{1/3} \). The goal is to separate variables, getting all \( y \) terms on one side and \( t \) (or derivatives of \( t \)) on the other.

Step 2: Separate Variables

Rewrite the equation as \( y^{1/3} dy = dt \). This allows us to integrate both sides separately.

Step 3: Integrate Both Sides

Integrate the left side with respect to \( y \) and the right side with respect to \( t \):\[ \int y^{1/3} dy = \int dt \]The integrals are: \( \frac{3}{4}y^{4/3} = t + C \), where \( C \) is the integration constant.

Step 4: Apply the Initial Condition

Use the initial condition \( y(0) = 0 \). Substitute \( t = 0 \) and \( y = 0 \) into the integrated equation:\[ \frac{3}{4}(0)^{4/3} = 0 + C \]This simplifies to \( C = 0 \).

Step 5: Solve for y

Substitute \( C = 0 \) back into \( \frac{3}{4}y^{4/3} = t \) and solve for \( y \):\[ y^{4/3} = \frac{4}{3}t \]Raising both sides to the power of \( 3/4 \), we get:\[ y = \left(\frac{4}{3}t\right)^{3/4} \]

Step 6: Verify Non-Constant Solution

The solution \( y = \left(\frac{4}{3}t\right)^{3/4} \) is non-constant and satisfies the differential equation. Additionally, since \( y(0) = 0 \), it meets the initial condition. The constant solution \( y = 0 \) is another valid solution, demonstrating the non-uniqueness of solutions.

Key Concepts

Differential Equation

A differential equation is an equation that involves derivatives of a function. In simpler terms, it's an equation that relates a function with its rate of change. These equations are fundamental in describing various physical, biological, and economic phenomena.

In the original exercise, the differential equation is given as \( y' = y^{1/3} \). Here, \( y' \) denotes the derivative of \( y \) with respect to time \( t \). To solve such equations, one often looks for functions that satisfy the equation along with any initial conditions that may be given.

Initial Value Problem

An initial value problem is a differential equation coupled with a specific value, known as the initial condition. This initial condition specifies the value of the function at a particular point, which helps in finding a unique solution to the problem.

• In our exercise, the initial condition is \( y(0) = 0 \). This means that at time \( t = 0 \), the value of \( y \) is 0.

By applying the initial condition after solving the differential equation, we ensure that the particular solution satisfies all aspects of the problem. This is crucial for ensuring that the solution is not just any function but one that meets the specific needs of the problem described.

Integration Constants

When solving a differential equation through integration, integration constants are introduced. These constants arise because the process of taking an antiderivative or integrating a function is not uniquely reversible. In other words, multiple functions can have the same derivative, differing only by a constant.

In our solution, after integrating, we obtain \( \frac{3}{4}y^{4/3} = t + C \), where \( C \) is the integration constant. This constant can be any real number, and its value is determined by substituting the initial condition into the equation.

• For this specific problem, using \( y(0) = 0 \) resulted in \( C = 0 \).

Understanding integration constants is important as they highlight the flexibility in finding solutions, allowing us to adjust the solution to meet initial or boundary conditions.

Non-uniqueness of Solutions

Differential equations can sometimes have more than one solution that satisfies both the equation and the initial conditions. This is referred to as non-uniqueness.

In the given exercise, while solving the problem, we find a non-constant solution: \( y = \left(\frac{4}{3}t\right)^{3/4} \), which satisfies both the differential equation and the initial condition \( y(0) = 0 \). However, there is also a constant solution: \( y(t) = 0 \). Both are valid solutions to the initial value problem.

• This illustrates that certain problems allow for multiple solutions, revealing the underlying complexity and richness of differential equations.

Non-uniqueness challenges the assumption that every problem has a single clean solution, encouraging broader thinking and further exploration of possible solutions.