Problem 29 - Calculus

Question

Explain what is wrong with the statement. \(Q=6 e^{4 t}\) is the general solution to the differential equation \(d Q / d t=4 Q.\)

Step-by-Step Solution

Answer

The statement is incorrect because the general solution should be \(Q = Ce^{4t}\), not \(Q = 6 e^{4t}\), to allow for any constant \(C\).

Step 1: Understand the Problem

The task is to determine if the given function \(Q = 6 e^{4t}\) is indeed a general solution to the differential equation \(\frac{dQ}{dt} = 4Q\). The general solution must solve the differential equation for any initial condition.

Step 2: Verify if the Proposed Solution Solves the Differential Equation

To verify, calculate the derivative of \(Q = 6 e^{4t}\). This gives \(\frac{dQ}{dt} = 6 \cdot 4 e^{4t} = 24 e^{4t}\). Compare this with \(4Q\) where \(Q = 6 e^{4t}\). Thus, \(4Q = 4 \times 6 e^{4t} = 24 e^{4t}\). Both expressions are equal, so \(Q = 6 e^{4t}\) satisfies the differential equation.

Step 3: Determine if it Represents the General Solution

The solution to the differential equation \(\frac{dQ}{dt} = 4Q\) is of the form \(Q = Ce^{4t}\) where \(C\) is a constant. The expression \(6 e^{4t}\) represents a specific solution with \(C = 6\), not the general solution for a variable constant \(C\).

Step 4: Identify the Error

The statement that \(Q = 6 e^{4t}\) is the general solution is incorrect because a general solution should contain an arbitrary constant \(C\), allowing for any initial value condition, not a specific constant like 6.

Key Concepts

General Solution

In differential equations, the **general solution** refers to the most comprehensive solution that accounts for all possible initial conditions. In simple terms, it's the solution that includes an arbitrary constant, allowing the function to take on multiple forms based on different initial conditions.

Consider the differential equation \(\frac{dQ}{dt} = 4Q\). Its general solution is \(Q = Ce^{4t}\), where \(C\) is an arbitrary constant. This means that depending on the initial value condition applied, \(C\) can adopt various values to fully describe the behavior of \(Q\) over time.

The key takeaway here is the flexibility provided by the arbitrary constant \(C\):

• Allows the solution to fit different starting points.
• Ensures the equation can describe any particular solution depending on the initial conditions specified.

In contrast, the statement claiming \(Q = 6e^{4t}\) is a general solution is incorrect, as it represents a specific solution when \(C\) is fixed at 6.

Initial Condition

An **initial condition** in the context of differential equations is a boundary that specifies the value of the solution at a particular point in time. Providing an initial condition helps in determining a particular solution from the general solution, which usually contains arbitrary constants.

Suppose we have the general solution \(Q = Ce^{4t}\). To find a specific solution, we might be given an initial condition like \(Q(0) = 6\).

• Substitute \(t = 0\) into the general form: \(Q(0) = Ce^{4 \cdot 0} = 6\).
• This simplifies to \(C = 6\).

Thus, the initial condition \(Q(0) = 6\) helps us find that specific solution, which is \(Q = 6e^{4t}\). This is now fixed and valid only under that initial condition, unlike the general solution which remains versatile for different initial conditions.

Arbitrary Constant

The **arbitrary constant** \(C\) in the general solution of a differential equation is crucial for encompassing all potential solutions of the equation simultaneously. It represents the unknowns that can fit various initial conditions.

Taking the example of the differential equation \(\frac{dQ}{dt} = 4Q\), its general solution is expressed as \(Q = Ce^{4t}\). Here, \(C\) is considered arbitrary because it permits:

• The solution to adapt to various initial starting points \(Q(0)\).
• Flexibility for the solution to encapsulate all possible trajectories defined by different initial conditions.

When no initial condition is provided, \(C\) remains open, underscoring its arbitrary nature and utility in solutions.

Thus, the error in stating \(Q = 6e^{4t}\) as the general solution lies in the assumption that \(C = 6\) solves all cases, while it actually limits the solution to a specific scenario.