Question

Here is a different example of the disagreeable things that can happen with nonlinear equations. Consider the nonlinear equation \(\dot{x}=2 \sqrt{x} .\) since this is first-order, one would expect that specification of \(x(0)\) would determine a unique solution. Show that for this equation there are two different solutions, both satisfying the initial condition \(x(0)=0 .\) [Hint: Find one solution \(x_{1}(t)\) by separating variables, but note that \(x_{2}(t)=0\) is another. Fortunately none of the equations normally encountered in classical mechanics suffer from this disagreeable ambiguity.]

Short answer

There are two solutions: \( x_1(t) = t^2 \) and \( x_2(t) = 0 \).

Step-by-step solution

Separate Variables

The given equation is \( \dot{x} = 2 \sqrt{x} \). Separate variables to rewrite this as \( \frac{dx}{\sqrt{x}} = 2dt \). This allows us to integrate on both sides.

Integrate Both Sides

Integrate both sides: \( \int \frac{dx}{\sqrt{x}} = \int 2dt \). The left side integrates to \( 2\sqrt{x} \), and the right side integrates to \( 2t + C \) where \( C \) is an integration constant. So, we have: \( 2\sqrt{x} = 2t + C \).

Solve for \( x \)

Solve for \( x \) from \( 2\sqrt{x} = 2t + C \), resulting in \( \sqrt{x} = t + \frac{C}{2} \), and squaring both sides gives \( x = \left( t + \frac{C}{2} \right)^2 \).

Apply Initial Condition

Apply the initial condition \( x(0)=0 \): \( 0 = \left( 0 + \frac{C}{2} \right)^2 \), which implies \( C = 0 \). Thus, one solution is \( x_1(t) = t^2 \).

Verify Alternate Solution

Note that the constant solution \( x_2(t) = 0 \) also satisfies \( \dot{x} = 2\sqrt{x} \) as \( 2\sqrt{0} = 0 \), and the initial condition \( x(0)=0 \).

Discuss Non-Uniqueness

Since both \( x_1(t) = t^2 \) and \( x_2(t) = 0 \) satisfy the differential equation and the initial condition, the problem demonstrates non-uniqueness for the given nonlinear equation.

Key concepts

Initial Value Problem

An Initial Value Problem (IVP) involves finding a function that satisfies a differential equation and meets a specified value at a given point, often expressed as an initial condition. In the context of our nonlinear equation, \[ \dot{x} = 2 \sqrt{x}, \]the initial condition is given by \( x(0) = 0 \).
This means that when time \( t \) is zero, the value of the function \( x(t) \) should be zero.

This type of problem is critical in mathematical modelling because it provides a starting point from which the behavior of a system can be predicted or explained. The initial condition is crucial as it can often dictate the nature of the solution, especially when dealing with nonlinear differential equations like in this example.

• The initial value constrains the range of potential solutions.
• Solving an IVP usually involves finding constants introduced during integration.

In our exercise, we see that it challenges the uniqueness of the solution, emphasizing that the initial condition doesn't always guarantee a single solution.

Separation of Variables

Separation of Variables is a fundamental technique used to solve differential equations, particularly when the equation allows the variables to be "separated" onto different sides of the equation. This method makes integration straightforward.
In our case with \[ \dot{x} = 2 \sqrt{x}, \]we separated variables to get:\[\frac{dx}{\sqrt{x}} = 2dt.\]
Once separated, we can integrate both sides independently:
- The left side \( \int \frac{dx}{\sqrt{x}} \) becomes \( 2 \sqrt{x} \).
- The right side \( \int 2dt \) evaluates to \( 2t + C \), where \( C \) is an arbitrary constant of integration.

• We isolate the variable \( x \) in the differential equation.
• Each side is integrated with respect to its own variable.
• Introduced constant \( C \) helps adjust the equation based on initial conditions.

This step is essential as it transforms a potentially complex problem into a simpler one with standard integration techniques.

Solution Uniqueness

Solution Uniqueness refers to when a differential equation has precisely one solution that satisfies the initial condition. Typically, solving an Initial Value Problem (IVP) for an ordinary differential equation yields a unique solution.
However, in our example, the nonlinear nature causes ambiguity.

• Function \( x_1(t) = t^2 \) satisfies the equation \( \dot{x} = 2 \sqrt{x} \) with \( x(0) = 0 \).
• Alternative solution \( x_2(t) = 0 \) also satisfies both the equation and initial condition.

This dual solution highlights non-uniqueness, a common occurrence in nonlinear equations. Non-uniqueness can arise in specific conditions:

- When multiple functions satisfy the same equation and initial condition.
- When the method of separation does not incorporate enough restricting conditions to filter out multiple possible solutions.

Understanding these nuances is crucial, especially in practical applications where predicting a single outcome might be critical.

Classical Mechanics

Classical Mechanics is the field of physics dealing with the motion of macroscopic objects. It often involves applying various mathematical models to predict and describe physical systems.
In classical mechanics, the equations of motion—often nonlinear differential equations—need to have unique solutions to effectively model real-world phenomena.

• Equations typically encounter issues of non-uniqueness when not associated with physical systems or when improperly constrained.
• The example with \( \dot{x} = 2 \sqrt{x} \)is more a mathematical curiosity than a physical model in classical mechanics.

Generally, equations derived from Newton's laws of motion and other classical models are structured to avoid this ambiguity.
This ensures that the solutions reliably predict and describe movements, meeting both initial conditions and natural physical constraints like energy conservation and momentum. Understanding differential equations in classical mechanics provides insight into why unique solutions are preferred and expected.